Quadratic Equations is a chapter in the CBSE Class 10 Mathematics syllabus from Mathematics. This chapter hub brings together revision notes, practice questions, worksheets, flashcards, formula sheet to help students learn, practice, and revise Quadratic Equations effectively.

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Quadratic Equations

Q: What is a quadratic equation?

A quadratic equation is a polynomial equation of degree 2, typically expressed in the form ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. It represents a parabola when graphed.

Q: How can we identify a quadratic equation?

You can identify a quadratic equation by checking if the highest power of the variable x is 2. If the equation can be written in the form ax^2 + bx + c = 0, it qualifies as a quadratic equation.

Q: What are the different methods to solve quadratic equations?

Quadratic equations can be solved using several methods including factorization, completing the square, and using the quadratic formula x = [-b ± √(b²-4ac)] / 2a, where a, b, and c are the coefficients in the equation.

Q: What is the purpose of the quadratic formula?

The quadratic formula provides a standardized method to find the roots of any quadratic equation ax^2 + bx + c = 0. It accounts for all cases: two real roots, one real root, or two complex roots.

Q: What are the roots of a quadratic equation?

Roots of a quadratic equation are the values of x that satisfy the equation, making it equal to zero. They can be real or complex numbers depending on the discriminant (b² - 4ac) of the equation.

Q: What is the discriminant in a quadratic equation?

The discriminant, denoted as D = b² - 4ac in the quadratic equation ax^2 + bx + c = 0, determines the nature of the roots: if D > 0, there are two distinct real roots; if D = 0, there is one real root; and if D < 0, the roots are complex.

Q: Can all quadratic equations be factored?

Not all quadratic equations can be factored easily into integer factors. However, if the discriminant is a perfect square, the equation can usually be factored over the integers.

Q: What real-life applications do quadratic equations have?

Quadratic equations are used in various real-life applications including physics (projectile motion), economics (profit maximization), and engineering (design of structures). They model situations where relationships can be quadratic.

Q: What is the historical significance of quadratic equations?

Quadratic equations have a rich history, with early solutions traced back to the Babylonians, Greeks, and Indian mathematicians like Brahmagupta, who developed methods still relevant today in solving these equations.

Q: How do you complete the square for a quadratic equation?

To complete the square for the equation ax^2 + bx + c = 0, you isolate the constant term, then manipulate the equation to express it as a perfect square trinomial, allowing for easier root finding.

NCERT Class 10 Mathematics Chapter 4: Quadratic Equations (Pages 38–48)

Class 10 CBSE hub Mathematics chapters Mathematics

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Summary of Quadratic Equations

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Quadratic Equations at a Glance

Board

CBSE

Class

Class 10

Subject

Mathematics

Book

Mathematics

Chapter

Pages

38–48

Resources

5 study resources

Quadratic Equations Summary

In this chapter, you will explore quadratic equations, which are fundamental in many real-world situations. A quadratic equation arises when a polynomial of the form ax squared plus bx plus c is set equal to zero, where a is not equal to zero. This equation can help solve problems related to areas, shapes, and various behaviors in the natural world. For example, consider a problem where a charity wishes to build a prayer hall. If the hall has an area of three hundred square meters and its length is one meter more than twice its breadth, we can form a quadratic equation. By assigning the breadth of the hall as x meters, we can express the length as two times x plus one meter. Hence, the area calculation would lead us to set up the equation two x squared plus x minus three hundred equals zero. This forms a quadratic equation that can be solved to determine the dimensions of the hall. Historically, the roots of quadratic equations have been studied for centuries. The Babylonians were among the first to solve quadratic equations, focusing on finding two positive numbers with a specific sum and product. This challenge mirrors the current approach to solving equations of the form x squared minus px plus q equals zero. Additionally, ancient Greek mathematicians like Euclid utilized geometric methods to find lengths relevant to these equations. The development of solving quadratic equations is often attributed to ancient Indian mathematicians. Brahmagupta provided an explicit formula for equations shaped like ax squared plus bx equals c. His advancements laid the groundwork for later mathematicians, such as Sridharacharya, who introduced a method known as completing the square, leading to the quadratic formula we use today. This formula effectively allows us to find solutions to quadratic equations. Scholars like Al-Khwarizmi also contributed to the understanding of quadratic equations, and during European medieval times, Abraham bar Hiyya Ha-Nasi published comprehensive solutions to various forms of these equations. The significance of quadratic equations persists into modern mathematics, as they apply to a broad array of real-life scenarios, including physics, engineering, and finance. As you delve into this chapter, you will learn various methods for finding the roots of quadratic equations, such as factoring, using the quadratic formula, and graphing. Additionally, you will explore how these equations can model real-life situations, reinforcing their relevance and usefulness in problem-solving. By comprehending quadratic equations, you will gain essential skills for advanced mathematics and everyday situations.

Quadratic Equations Revision Guide

Download the Quadratic Equations revision guide with key points, summaries, and quick revision notes for CBSE Class 10 Mathematics.

Key Points

Quadratic equation definition

A quadratic equation in one variable is any equation reducible to ax^2 + bx + c = 0 with a, b, c real and a ≠ 0.

Meaning of coefficients

In ax^2+bx+c=0: a is coefficient of x^2, b of x, and c is the constant term.

Standard form is essential

Expand and simplify, then bring all terms to one side so RHS becomes 0; then identify a, b, c correctly.

Roots / solutions

Roots are values of x that make the equation true (LHS becomes 0). A quadratic can have 0, 1, or 2 real roots.

Zeros of polynomial connection

Roots of ax^2+bx+c=0 are the same as zeros of p(x)=ax^2+bx+c because solving means finding x where p(x)=0.

Modelling from situations

Choose a variable for an unknown, express other quantities in terms of it, use the given condition (area/product/etc.), and simplify to a quadratic.

Area-based quadratics

Rectangle area L×B often becomes (expression)(expression)=given value, leading to a quadratic after expansion.

Example structure (hall/plot)

If breadth=x and length=(2x+1), area condition x(2x+1)=300 gives 2x^2+x−300=0.

Context constraints

For lengths and many physical quantities, x>0. After solving, reject roots that violate the context.

Factorisation method overview

Convert ax^2+bx+c to a product of two linear factors and then solve using the zero product rule.

Zero product rule

If AB=0, then A=0 or B=0. This converts one quadratic equation into two linear equations.

Factorising when a=1

For x^2+bx+c, find m,n such that m+n=b and mn=c, then write (x+m)(x+n)=0.

Factorising when a≠1

Use splitting the middle term: find two numbers with product a·c and sum b; split bx and factor by grouping.

Factor by grouping steps

After splitting: group terms in pairs, take common factors, then factor out the common binomial to get two factors.

Checking roots

Substitute each root back into the original equation (or situation) to confirm the LHS becomes 0 and conditions hold.

Discriminant definition

For ax^2+bx+c=0, discriminant D=b^2−4ac. It predicts the nature of roots.

Nature of roots by D

D>0: two distinct real roots; D=0: one real repeated root; D<0: no real roots.

Graph interpretation

D>0 means parabola cuts x-axis twice; D=0 touches x-axis once; D<0 does not intersect x-axis.

Perfect square discriminant

If D is a perfect square (and a,b,c are integers), then √D is rational and roots are rational numbers.

Quadratic formula (for completeness)

x=(−b±√D)/(2a). It works for every quadratic with a≠0 and uses the same discriminant D.

Common errors in D

Square b correctly (sign disappears), and remember D has minus 4ac; be careful when c is negative.

Common factor special case

If c=0, then ax^2+bx= x(ax+b)=0, giving one root x=0 immediately.

Difference of squares

x^2−k^2 factors as (x−k)(x+k), a frequent quick factorisation pattern.

Perfect square trinomials

x^2±2kx+k^2 factors as (x±k)^2 and has equal roots because D=0.

Method selection in exams

Use factorisation when factors are simple; use discriminant when asked about nature; use formula when factorisation is hard.

Answer-writing checklist

Write standard form, show factorisation/steps, state both roots clearly, and verify roots (and context, if given).

Quadratic Equations Practice Questions & Answers

Practice important questions and exam-style problems from Quadratic Equations. These questions cover key topics from the CBSE Class 10 Mathematics syllabus.

How to practice: Start with the questions below to test your understanding of Quadratic Equations. Use the revision guide to review concepts you find difficult, then come back and retry the questions for better retention.

What is the standard form of a quadratic equation?

Single Answer MCQ

Q-00173828

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In the equation 2x^2 + 3x - 5 = 0, what is the value of 'a'?

Single Answer MCQ

Q-00173829

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What are the roots of the quadratic equation x^2 - 6x + 9 = 0?

Single Answer MCQ

Q-00173830

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Which of the following describes the vertex of a parabola represented by a quadratic equation?

Single Answer MCQ

Q-00173831

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If a quadratic equation has two equal roots, what can you infer about its discriminant?

Single Answer MCQ

Q-00173832

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Identify the maximum number of distinct real roots a quadratic equation can have.

Single Answer MCQ

Q-00173833

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The area of a rectangular garden is represented by the equation x^2 - 7x + 10. What are the dimensions of the garden?

Single Answer MCQ

Q-00173834

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What value of 'b' gives the vertex for the equation y = x^2 + bx + 8 at the lowest point?

Single Answer MCQ

Q-00173835

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View all 176 Quadratic Equations questions

What does the graph of the equation y = -x^2 + 4 represent?

If the roots of a quadratic equation are p and q, what relation does (p+q) have with the coefficients?

For which of the following values of 'c' will the equation x^2 + 4x + c = 0 have two distinct real roots?

What x-value gives the minimum for the function y = 2x^2 - 4x + 1?

Which method is commonly used to derive the quadratic formula?

In a quadratic equation of the form ax^2 + bx + c = 0, if a = 0, what happens?

What is the factorization of the quadratic equation x^2 - 5x + 6?

If a quadratic equation x^2 + 7x + 10 = 0 is solved by factorization, what are the roots?

Which equation represents the factorization of x^2 - 9?

Factor the quadratic equation: x^2 + 8x + 15 = 0.

The quadratic equation x^2 - 4x - 12 = 0 can be factored as which of the following?

To solve the equation x^2 + 5x + 6 = 0 by factorization, which factors should be identified?

What should the factor pair of x^2 - x - 6 look like?

In the quadratic equation 2x^2 - 8x = 0, what is the first step when using factorization?

For the quadratic equation x^2 - 7x + 10 = 0, which is a true statement about the factorization?

Which form shows the correct procedure to factor x^2 + 2x - 8?

When factorizing the quadratic equation 3x^2 + 12x = 0, what is first required?

What is the solution for the roots of x^2 - 6x + 8 = 0?

In the equation x^2 - 12 = 0, what factorization is correct?

To factor the quadratic x^2 + 16x + 64, what method will yield the correct roots?

How do you factor the equation x^2 - 14x + 49 = 0?

What remains true for the roots of a quadratic equation ax^2 + bx + c = 0 if the discriminant is positive?

For which value of the discriminant do the quadratic equation's roots become equal?

Which of the following statements is true regarding the roots of the equation x^2 - 6x + 9 = 0?

What can be inferred if a quadratic equation has roots that are both negative?

Given the quadratic equation 2x^2 + 8x + 8 = 0, what are the roots of this equation?

What is the nature of the roots for the equation 3x^2 + 2x + 1 = 0?

If the roots of a quadratic equation are 5 and 3, what is the sum of the roots?

In a quadratic equation ax^2 + bx + c, if a is negative, what can be inferred about the direction of the parabola?

A quadratic equation has a positive discriminant. This indicates what about the roots?

What happens if the quadratic equation's roots are imaginary?

What does a discriminant of 25 signify for a quadratic equation?

If a quadratic equation has roots that are both positive, what do we know about the coefficients?

For the equation x^2 + 4x + 4 = 0, how many distinct roots does it have?

If a quadratic equation has roots represented as α and β, what is the formula for the sum of the roots?

What is the relationship between the discriminant and the nature of roots if it is less than zero?

What is the standard form of a quadratic equation?

If a quadratic equation has roots p and q, what is the sum of the roots?

What does it mean if the discriminant of a quadratic equation is zero?

Which of the following equations is not a quadratic equation?

In the context of real-life applications, which scenario can be modeled using a quadratic equation?

Which method can be used to solve the quadratic equation ax^2 + bx + c = 0?

What is the quadratic formula used to find the roots of ax^2 + bx + c = 0?

How many real roots does the quadratic equation 2x^2 + x - 300 = 0 have?

Which of the following statements about quadratic equations is true?

In the quadratic function f(x) = ax^2 + bx + c, what effect does the coefficient 'a' have?

Given the equation 4x^2 - 12x + 9 = 0, what type of roots does it have?

What real-life application does the quadratic equation x^2 - 5x + 6 = 0 represent?

Three consecutive positive integers are such that sum of square of the first and the product of the other two is 67, find the integers.

The value of k for which the equation 2kx^2 - 6x + 3 = 0 has real and equal roots, is:

Observe the given figure showing six identical rectangular enclosures made by using fencing wire mesh. Dimensions of each enclosure is x feet × y feet. The total length of fencing required is 152 feet and area of each enclosure is 80 square feet. Write an expression for length of fencing required in terms of x and y.

Using the above equation in quadratic form, solve the equation and find the dimensions of each enclosure using quadratic formula.

Write the above equation in quadratic equation form and thus find the dimensions of each enclosure using factorisation method.

For the six identical rectangular enclosures, write the area of each enclosure in terms of x.

A faster train takes one hour less than a slower train for a journey of 200 km. If the speed of the slower train is 10 km/hr less than that of the faster train, find the speeds of the two trains.

The sum of the areas of two squares is 640 m^2. If the difference in their perimeters is 64 m, find the sides of the two squares.

The discriminant of the quadratic equation x^2 - 3x - 2 = 0 is:

The equation x + 1/x = 3 (x ≠ 0) is expressed as a quadratic equation in the form of ax^2 + bx + c = 0. The value of a - b + c is:

The difference of the squares of two positive numbers is 180. The square of the smaller number is 8 times the greater number. Find the two numbers.

Find the value(s) of k for which the equation 2x^2 + kx + 3 = 0 has real and equal roots. Hence, find the roots of the equations so obtained.

The value of m for which the quadratic equation 3x² − 7x + m = 0 has real and equal roots, is

The value of p for which roots of the quadratic equation x² − px + 6 = 0 are rational, is

Two water taps together can fill a tank in 8 8/9 hours. The tap of larger diameter takes 4 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

A person on a tour has ₹4,200 for expenses. If he extends his tour for 3 days, he has to cut down his daily expenses by ₹70. Find the original duration of the tour.

The area of a right-angled triangle is 600 cm^2. If the base of the triangle exceeds the altitude by 10 cm, find all the three dimensions of the triangle.

In a flight of 600 km, an aircraft slowed down its speed due to bad weather. Its average speed for the trip reduced by 200 km/h from its usual speed and time of flight increased by 30 minutes. Find the scheduled duration of the flight.

Two pipes are used to fill a swimming pool. If the pipe of the larger diameter is used for 4 hours and the pipe of the smaller diameter for 9 hours, only half of the pool can be filled. Find how long it would take for each pipe to fill the pool, separately, if the pipe of smaller diameter takes 10 hours more than the pipe of larger diameter to fill the pool.

In a class test, the sum of Anamika’s marks obtained in Maths and Science is 30. Had she got 2 marks more in Maths and 3 marks less in Science, the product of the marks would have been 210. Find the marks she got in the two subjects.

The length of hypotenuse (in cm) of a right-angled triangle is 6 cm more than twice the length of its shortest side. If the length of its third side is 6 cm less than thrice the length of its shortest side, find the dimensions of the triangle.

The roots of the quadratic equation (x − 1)^2 = 16 are:

If the roots of the quadratic equation √3 x^2 − kx + 2√3 = 0 are real and equal, then the value(s) of k is/are:

Find two consecutive negative integers, sum of whose squares is 481.

If the roots of the quadratic equation √3x² – kx + 2√3 = 0 are real and equal, then the value(s) of k is/are:

The roots of the quadratic equation 4x² – (a – 1)² = 0 are:

Find two consecutive negative integers, sum of whose squares is 481.

If the roots of the quadratic equation √3x^2 - kx + 2√3 = 0 are real and equal, then the value(s) of k is/are:

The roots of the quadratic equation (x - 1)^2 = 16 are:

Find two consecutive negative integers, sum of whose squares is 481.

A faster train takes one hour less than a slower train for a journey of 200 km. If the speed of the slower train is 10 km/hr less than that of the faster train, find the speeds of the two trains.

The sum of the areas of two squares is 640 m². If the difference in their perimeters is 64 m, find the sides of the two squares.

A faster train takes one hour less than a slower train for a journey of 200 km. If the speed of the slower train is 10 km/hr less than that of the faster train, find the speeds of the two trains.

The sum of the areas of two squares is 640 m². If the difference in their perimeters is 64 m, find the sides of the two squares.

The discriminant of the quadratic equation –x² – 5x + 6 = 0 is:

The equation x + 1/x = 3 (x ≠ 0) is expressed as a quadratic equation in the form ax² + bx + c = 0. The value of a – b + c is:

Find the value(s) of k for which the equation 2x² + kx + 3 = 0 has real and equal roots. Hence, find the roots of the equations so obtained.

The difference of the squares of two positive numbers is 180. The square of the smaller number is 8 times the greater number. Find the two numbers.

The discriminant of the quadratic equation 2x^2 - 3x - 5 = 0 is:

The equation x + 1/x = 3 (x ≠ 0) is expressed as a quadratic equation in the form ax^2 + bx + c = 0. The value of a - b + c is:

Find the value(s) of k for which the equation 2x^2 + kx + 3 = 0 has real and equal roots. Hence, find the roots of the equations so obtained.

The difference of the squares of two positive numbers is 180. The square of the smaller number is 8 times the greater number. Find the two numbers.

The value of ‘a’ for which ax² + 3x + 1 = 0 has real and equal roots is:

Which of the following equations is a quadratic equation?

The sum of areas of two squares is 2650 cm². If the sum of their perimeters is 280 cm, find the sides of the two given squares.

Express the equation 1/x − 1/(x − 2) = 3, x ≠ 0, 2, as a quadratic equation in standard form. Hence, find the roots of the quadratic equation so obtained.

The value of a for which x² + 3x + a = 0 has real and equal roots is:

Which of the following equations is a quadratic equation?

The sum of areas of two squares is 2650 cm². If the sum of their perimeters is 280 cm, find the sides of the two given squares.

Express the equation 1/x - 1/(x - 2) = 3, x ≠ 0, 2, as a quadratic equation in standard form. Hence, find the roots of the quadratic equation so obtained.

One of the values of p for which px² + 4x + p = 0 has real and equal roots is:

Which of the following equations is a quadratic equation?

The sum of areas of two squares is 2650 cm². If the sum of their perimeters is 280 cm, find the sides of the two given squares.

Express the equation 1/x − 1/(x − 2) = 3, where x ≠ 0, 2, as a quadratic equation in standard form. Hence, find the roots of the quadratic equation so obtained.

If x = √x, (x ≠ 0) is expressed as a quadratic equation in the form ax² + bx + c = 0, then the value of a + b + c is:

Assertion (A): Every quadratic equation has two real roots. Reason (R): A quadratic polynomial can have at most two zeroes.

Find two consecutive odd integers, sum of whose squares is 290.

A charity trust decides to build a rectangular hall having an area of 300 m². The length of the hall is one metre more than twice its width. Find the length and breadth of the hall.

If (√x + 1)² = x² + 2√x is expressed as a quadratic equation in the form ax² + bx + c = 0, then the value of a – b + c is:

Assertion (A): Every quadratic equation has two real roots. Reason (R): A quadratic polynomial can have at most two zeroes.

Find two consecutive odd integers, sum of whose squares is 290.

A charity trust decides to build a rectangular hall having an area of 300 m². The length of the hall is one metre more than twice its width. Find the length and breadth of the hall.

If (x + 1)^3 = x^3 + 1 is expressed as a quadratic equation in the form px^2 + qx + r = 0, then the value of p − q + r is:

Assertion (A): Every quadratic equation has two real roots. Reason (R): A quadratic polynomial can have at most two zeroes.

Find two consecutive odd integers, sum of whose squares is 290.

A charity trust decides to build a rectangular hall having an area of 300 m². The length of the hall is one metre more than twice its width. Find the length and breadth of the hall.

The value of k for which the equation 2kx^2 - 6x + 3 = 0 has real and equal roots, is:

Observe the figure showing six identical rectangular enclosures of dimensions x feet × y feet made using fencing wire mesh. The total length of fencing required is 152 feet and area of each enclosure is 80 square feet. Write an expression for length of fencing required in terms of x and y.

Observe the figure showing six identical rectangular enclosures of dimensions x feet × y feet. Write the area of each enclosure in terms of x.

Using the above equation in quadratic form, solve the equation and find the dimensions of each enclosure using quadratic formula.

Write the above equation in quadratic equation form and thus find the dimensions of each enclosure using factorisation method.

The value of k for which the equation 5x^2 - 2x + k = 0 has equal real roots, is:

Six identical rectangular enclosures are arranged as shown in the figure. Dimensions of each enclosure are x feet by y feet. Write an expression for length of fencing required in terms of x and y.

Each enclosure has dimensions x feet by y feet and area 80 square feet. Write the area of each enclosure in terms of x.

For the six identical rectangular enclosures, the total fencing required is 152 feet and area of each enclosure is 80 square feet. Write the equation in quadratic equation form and find the dimensions of each enclosure using factorisation method.

For the six identical rectangular enclosures, using the equation in quadratic form, solve the equation and find the dimensions of each enclosure using quadratic formula.

The roots of the quadratic equation x^2 + 9 = 0 are

ABCD is a rectangle of dimensions 80 cm × 60 cm. Another rectangle PQRS is drawn inside ABCD leaving space of equal width x cm along the edges of ABCD. If area PQRS is half of the area ABCD, then find the value of x.

A train covers a distance of 90 km at a uniform speed. Had the speed been 15 km/h more, it would have taken 30 minutes less for the same journey. Find the original speed of the train.

The roots of the quadratic equation x^2 + 9 = 0 are

A train covers a distance of 90 km at a uniform speed. Had the speed been 15 km/h more, it would have taken 30 minutes less for the same journey. Find the original speed of the train.

The roots of the quadratic equation x² + 9 = 0 are

A train covers a distance of 90 km at a uniform speed. Had the speed been 15 km/h more, it would have taken 30 minutes less for the same journey. Find the original speed of the train.

If x = 3 is a solution of the equation ax^2 + 3x – 12 = 0, then

Solve for x: (x – 2)/(x – 3) + (x – 4)/(x – 5) = 10/3; x ≠ 3, 5.

A motor boat, whose speed in still water is 20 km/h, takes 1 hour more to go 48 km upstream than to return downstream to the same point. Find the speed of the stream.

The value of m for which the quadratic equation 3x² - 7x + m = 0 has real and equal roots, is

The value of p for which roots of the quadratic equation x² - px + 6 = 0 are rational, is

The value of p for which roots of the quadratic equation x² – px + 6 = 0 are rational, is

The value of k for which the equation kx² – 6x – 4 = 0 has real and equal roots, is

By selling an article for ₹48, a trader loses as much percent as half of the cost price of the article. Calculate the cost price and loss amount of the article.

The value of k for which the equation kx^2 - 6x - 4 = 0 has real and equal roots, is

The value of p for which roots of the quadratic equation x^2 - px + 6 = 0 are rational, is

Venkat can row a boat in still water at the speed of 12 km/h. He ferries tourists 15 km upstream and 18 km downstream in 3 hours. Find the speed of the stream.

If the quadratic equation 9x^2 + 8kx + 16 = 0 has real and equal roots, then the value of k is

Verify that roots of the quadratic equation (p – q)x^2 + (q – r)x + (r – p) = 0 are equal when q + r = 2p.

A person on tour has ₹5,400 for his expenses. If he extends his tour by 5 days, he has to cut down his daily expenses by ₹180. Find the original duration of the tour and daily expense.

The total cost of certain piece of cloth was ₹2,100. During special sale time, the shopkeeper offered 2 m extra cloth for free thus reducing the price of cloth per metre by ₹120. What was the original per metre price of cloth and its length?

If roots of the quadratic equation x² - k√3 x + 2 = 0 are real and equal, then value of k is

Verify that roots of the quadratic equation (p - q)x² + (q - r)x + (r - p) = 0 are equal when q + r = 2p.

A person on tour has ₹5,400 for his expenses. If he extends his tour by 5 days, he has to cut down his daily expenses by ₹180. Find the original duration of the tour and daily expense.

The total cost of a certain piece of cloth was ₹2,100. During special sale time, the shopkeeper offered 2 m extra cloth for free, thus reducing the price of cloth per metre by ₹120. What was the original per metre price of cloth and its length?

If the quadratic equation 9x^2 + 8kx + 16 = 0 has real and equal roots, then the value of k is

Verify that roots of the quadratic equation (p − q)x^2 + (q − r)x + (r − p) = 0 are equal when q + r = 2p.

A person on tour has ₹5,400 for his expenses. If he extends his tour by 5 days, he has to cut down his daily expenses by ₹180. Find the original duration of the tour and daily expense.

Solve for x: 2x² - 2√2x + 1 = 0.

Find the value(s) of k for which the quadratic equation x² + 5kx + 16 = 0 has real and equal roots.

Case Study 1: Social work aims at fulfilment of human needs. Social workers aim to open the doors of access and opportunity for those who are in greatest need. Free education is a great social work. By doing so, we can remove illiteracy from our society. Rohan, being a social worker, wants to donate his land to the Village Panchayat for opening of a school. Rohan's land is in the form of a rectangle of dimensions 500 m × 400 m. The Village Panchayat decides to leave the area on all the four sides of the land for grass and flowers. If width of x m land is kept for grass and flowers on all the four sides as shown in Figure 4, find the lengths PQ and QR if area of grass and flowers region surrounding PQRS is 118400 m².

Case Study 1: For the rectangle PQRS formed inside Rohan's rectangular land of dimensions 500 m × 400 m, with width x m land kept for grass and flowers on all four sides as shown in Figure 4, also find the perimeter of the rectangle PQRS.

Solve for x: 2x^2 - 2√2 x + 1 = 0.

Find the value(s) of k for which the quadratic equation x^2 + 5kx + 16 = 0 has real and equal roots.

Case Study 2: Rohan's land is in the form of a rectangle of dimensions 500 m × 400 m. The Village Panchayat decides to leave area on all four sides of the land for grass and flowers. If width of x m land is kept for grass and flowers on all four sides as shown in Figure 3, find the lengths PQ and QR if the area of the grass and flowers region surrounding PQRS is 118400 m^2.

A two-digit number is such that the product of its digits is 12. When 36 is added to this number, the digits interchange their places. Find the number.

A student scored a total of 32 marks in class tests in Mathematics and Science. Had he scored 2 marks less in Science and 4 marks more in Mathematics, the product of his marks would have been 253. Find his marks in the two subjects.

What is the negative integer solution of the equation x^2 - 9 = 0?

Number

Q-00001187

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Quadratic Equations Practice Worksheets

Download and practice Quadratic Equations worksheets to improve problem-solving accuracy and speed for CBSE Class 10 Mathematics exams.

Questions

Write in standard form and identify a, b, c: x(x-4)=5.

x(x−4)=5 ⇒ x^2−4x=5 ⇒ x^2−4x−5=0. So a=1, b=−4, c=−5.

Form a quadratic equation: A rectangle has breadth x cm and length (x+3) cm. Area is 40 cm^2.

x(x+3)=40 ⇒ x^2+3x−40=0.

Solve by factorisation: x^2 - 9x + 20 = 0.

x^2−9x+20=(x−5)(x−4)=0 ⇒ x=5 or x=4.

Solve by factorisation: x^2 + 7x + 10 = 0.

(x+5)(x+2)=0 ⇒ x=−5 or x=−2.

Solve by taking common factor: 3x^2 - 12x = 0.

3x(x−4)=0 ⇒ x=0 or x=4.

Solve: 2x^2 - 7x + 3 = 0.

2x^2−7x+3=(2x−1)(x−3)=0 ⇒ x=1/2 or x=3.

From the area condition x(2x+1)=300, write the quadratic equation in standard form.

2x^2+x=300 ⇒ 2x^2+x−300=0.

Factorise and solve: 2x^2 + x - 300 = 0.

2x^2+x−300=2x^2+25x−24x−300= (x+12)(2x−25)=0 ⇒ x=−12 or x=25/2.

A length is represented by x. The equation gives roots x=25/2 and x=−12. Which value is acceptable and why?

Accept x=25/2 because a length must be positive. Reject x=−12 as it is not meaningful for length.

Compute the discriminant of x^2 - 5x + 6 = 0 and state the nature of roots.

D=25−24=1>0, so two distinct real roots.

Find k such that x^2 - 4x + k = 0 has equal roots.

D=16−4k=0 ⇒ k=4.

State the nature of roots for x^2 + 4x + 5 = 0 without solving.

D=16−20=−4<0, so no real roots.

For 2x^2 + x - 300 = 0, compute D and comment on rationality of roots.

a=2,b=1,c=−300 ⇒ D=1−4·2·(−300)=2401=49^2. D>0 so two distinct real roots, and since D is a perfect square, roots are rational.

Expand and simplify: (x-4)(x+1)=0 into standard form.

x^2 + x −4x −4=0 ⇒ x^2−3x−4=0.

Solve: x^2 - 9 = 0 and write both roots.

x^2−9=(x−3)(x+3)=0 ⇒ x=3 or x=−3.

A square has area 81 cm^2. Form the quadratic equation for side x and find x.

x^2−81=0 ⇒ (x−9)(x+9)=0 ⇒ x=9 or x=−9. Since side length is positive, x=9 cm.

Two consecutive integers have product 72. Form the quadratic equation for the smaller integer x.

x(x+1)=72 ⇒ x^2+x−72=0.

Solve the equation from the consecutive integer problem: x^2 + x - 72 = 0 and write the two integer solutions.

(x+9)(x−8)=0 ⇒ x=−9 or x=8. Integers are (−9,−8) or (8,9).

Solve: x^2 - 6x + 9 = 0 and interpret the result about roots.

(x−3)^2=0 ⇒ x=3 (repeated). Roots are real and equal.

A rectangle has perimeter 34 m. If breadth is x m and length is (x+4) m, find x.

2[(x+4)+x]=34 ⇒ 2(2x+4)=34 ⇒ 4x+8=34 ⇒ x=26/4=13/2 m. Then length = x+4 = 21/2 m.

Explain why dividing both sides of x(x-4)=0 by x can lose a solution.

x(x−4)=0 has roots x=0 and x=4. If we divide by x, we implicitly assume x≠0 and lose the root x=0.

Find the nature of roots of 3x^2 + 2x + 5 = 0.

D=2^2−4·3·5=4−60=−56<0, so no real roots.

A quadratic has discriminant 0. Write one possible example of such an equation and show D=0.

Example: x^2−6x+9=0. Here a=1,b=−6,c=9. D=36−36=0.

Write a short method-mark style solution outline for solving ax^2+bx+c=0 by factorisation (no example needed).

Outline: (1) Rewrite in standard form ax^2+bx+c=0. (2) Factorise into (px+q)(rx+s)=0. (3) Use AB=0 ⇒ px+q=0 or rx+s=0. (4) Solve both linear equations to get roots. (5) Substitute to check.

A rectangle has area 50 m^2 and length is 5 m more than breadth. Find breadth.

x(x+5)=50 ⇒ x^2+5x−50=0. Factorising: (x+10)(x−5)=0 ⇒ x=5 or x=−10. Breadth is positive, so x=5 m.

Without solving, decide if x^2 - 2x + 10 = 0 has real roots.

D=(−2)^2−4·1·10=4−40=−36<0, so no real roots.

Solve: 4x^2 - 12x = 0 and state the roots clearly.

4x(x−3)=0 ⇒ x=0 or x=3.

Quadratic Equations Formula Sheet

Use this Class 10 Mathematics Quadratic Equations Formula Sheet for quick revision before school exams and CBSE exams. It brings together the important formulas, key concepts, and worked examples in one place so students can revise faster and download a printable PDF for offline study.

Important Formulas

Standard Form of a Quadratic Equation: ax² + bx + c = 0

Here, a is the coefficient of x², b is the coefficient of x, and c is the constant. This is the fundamental form of a quadratic equation, used for either graphical representation or solution finding.

Quadratic Formula: x = (-b ± √(b² - 4ac)) / (2a)

This formula gives solutions for any quadratic equation in standard form. The discriminant (b² - 4ac) indicates the nature of roots: real and distinct, real and equal, or complex.

Factored Form: a(x - r₁)(x - r₂) = 0

Where r₁ and r₂ are the roots of the equation. Useful for quickly determining roots when given a product of factors.

Sum and Product of Roots: r₁ + r₂ = -b/a, r₁r₂ = c/a

These equalities relate the roots of the quadratic equations to coefficients. They simplify finding roots without full factorization.

Vertex Form: y = a(x - h)² + k

In this form, (h, k) is the vertex of the parabola represented by the quadratic equation. Useful for graphing and understanding the graph's maximum/minimum points.

Discriminant: D = b² - 4ac

D is used to determine the nature of the roots. If D > 0, roots are real and distinct; if D = 0, roots are real and equal; if D < 0, roots are complex.

Completing the Square: ax² + bx = k → (x + b/(2a))² = (b² - 4ac)/(4a)

This method transforms a quadratic into vertex form. It’s useful for deriving the quadratic formula and understanding the parabola.

Roots of Unity: x² - (r₁ + r₂)x + r₁r₂ = 0

This formulation shows how the sum and product of the roots relate to the coefficients, reaffirming connections between algebra and geometry.

Graph of a Quadratic: y = ax² + bx + c

The graph is a parabola, opening upwards (a > 0) or downwards (a < 0). Understanding this helps in sketching quadratic functions and analyzing their behavior.

Quadratic Inequality: ax² + bx + c > 0

This is used to find the intervals where a quadratic is positive/negative. It involves determining the roots and testing intervals.

Worked Examples

General Form: 2x² + x - 300 = 0

This particular equation is derived from a real-world scenario and can be solved using various methods, demonstrating practical applications of quadratics.

Example Quadratic Function: f(x) = x² - 5x + 6

This function can be analyzed to find its roots, vertex, and axis of symmetry, demonstrating the characteristics of its graph.

Factoring Example: x² - 7x + 10 = (x - 2)(x - 5) = 0

This shows how to factor a simple quadratic equation. Roots can be quickly identified as x = 2 or x = 5.

Graphical Representation: y = 2(x - 1)(x - 3)

Illustrates how to represent a quadratic equation in a factored manner, showing its roots clearly on a graph.

Inequality Example: x² - 4 < 0

This quadratic inequality can be solved to find intervals of x that satisfy the condition, enhancing critical thinking and problem-solving skills.

Using the Quadratic Formula: x = 4/3, -75/2 for 6x² + 5x + 4 = 0

An example using the quadratic formula to find non-integer solutions for a specific quadratic equation.

Area-Related Quadratic: x(x + 2) - 48 = 0

This equation arises from a real-world problem involving area dimensions, allowing for practical application during problem-solving.

Completing the Square: x² - 4x + 4 = 0 → (x - 2)² = 0

Shows how to transform and solve a quadratic equation by finding perfect square trinomials.

Vertex Calculation: V = (h, k) where h = -b/(2a), k = f(h)

Used to determine the vertex of the parabola, which assists in understanding its maximum/minimum value.

Real-life Application: x² + 8x + 16 = 0 → (x + 4)² = 0

Models a situation where the solution represents important dimensions or values in a context, reiterating the significance of quadratics.

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Quadratic Equations Frequently Asked Questions

Discover Quadratic Equations in Class 10 Mathematics. This chapter covers definitions, methods of solving, and real-world applications, enriching your understanding of this essential topic.

QWhat is a quadratic equation?

A quadratic equation is a polynomial equation of degree 2, typically expressed in the form ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. It represents a parabola when graphed.

QHow can we identify a quadratic equation?

You can identify a quadratic equation by checking if the highest power of the variable x is 2. If the equation can be written in the form ax^2 + bx + c = 0, it qualifies as a quadratic equation.

QWhat are the different methods to solve quadratic equations?

Quadratic equations can be solved using several methods including factorization, completing the square, and using the quadratic formula x = [-b ± √(b²-4ac)] / 2a, where a, b, and c are the coefficients in the equation.

QWhat is the purpose of the quadratic formula?

The quadratic formula provides a standardized method to find the roots of any quadratic equation ax^2 + bx + c = 0. It accounts for all cases: two real roots, one real root, or two complex roots.

QWhat are the roots of a quadratic equation?

Roots of a quadratic equation are the values of x that satisfy the equation, making it equal to zero. They can be real or complex numbers depending on the discriminant (b² - 4ac) of the equation.

QWhat is the discriminant in a quadratic equation?

The discriminant, denoted as D = b² - 4ac in the quadratic equation ax^2 + bx + c = 0, determines the nature of the roots: if D > 0, there are two distinct real roots; if D = 0, there is one real root; and if D < 0, the roots are complex.

QCan all quadratic equations be factored?

Not all quadratic equations can be factored easily into integer factors. However, if the discriminant is a perfect square, the equation can usually be factored over the integers.

QWhat real-life applications do quadratic equations have?

Quadratic equations are used in various real-life applications including physics (projectile motion), economics (profit maximization), and engineering (design of structures). They model situations where relationships can be quadratic.

QWhat is the historical significance of quadratic equations?

Quadratic equations have a rich history, with early solutions traced back to the Babylonians, Greeks, and Indian mathematicians like Brahmagupta, who developed methods still relevant today in solving these equations.

QHow do you complete the square for a quadratic equation?

To complete the square for the equation ax^2 + bx + c = 0, you isolate the constant term, then manipulate the equation to express it as a perfect square trinomial, allowing for easier root finding.

QWhat is meant by the nature of roots?

The term 'nature of roots' refers to the types of solutions a quadratic equation has. Based on the discriminant, roots can be real and distinct, real and equal, or complex.

QHow do you derive the quadratic formula?

The quadratic formula is derived from the process of completing the square on the standard quadratic equation ax^2 + bx + c = 0, systematically isolating x to find its values.

QWhat role does the coefficient 'a' play in a quadratic equation?

The coefficient 'a' in a quadratic equation ax^2 + bx + c determines the direction of the parabola: if 'a' is positive, the parabola opens upwards; if negative, it opens downwards.

QIn what scenarios would a quadratic equation have complex roots?

A quadratic equation has complex roots when the discriminant is less than zero (D < 0), indicating that the parabola does not intersect the x-axis.

QWho were the first to solve quadratic equations?

Many believe the Babylonians were the first to solve quadratic equations, having developed methods to find unknowns, which aligned with solving certain forms of quadratic equations.

QHow does a graph of a quadratic equation look?

The graph of a quadratic equation is a parabola. The vertex represents the maximum or minimum point, and the axis of symmetry divides the parabola into two mirror-image halves.

QWhat is a real-world example of using a quadratic equation?

A real-world example includes determining the maximum height of a thrown object. The object's trajectory can be modeled by a quadratic equation, allowing predictions about its peak position and flight time.

QCan quadratic equations have rational roots?

Yes, quadratic equations can have rational roots. If the discriminant is a perfect square, the quadratic formula yields rational solutions.

QHow is the sum of the roots of a quadratic equation related to its coefficients?

The sum of the roots of a quadratic equation ax^2 + bx + c = 0 is given by -b/a, while the product of the roots is given by c/a, forming the basis for Vieta's formulas.

QWhat is the significance of the vertex of a parabola?

The vertex of a parabola, represented in the vertex form of a quadratic equation, indicates the maximum or minimum point of the parabola, which is crucial in optimization problems.

QHow can quadratic equations be applied in finance?

In finance, quadratic equations can model situations such as profit maximization where revenue and cost functions are represented as quadratics, allowing for the determination of optimal investment levels.

QWhat method involves visualizing the graph to solve a quadratic equation?

Graphical methods involve plotting the quadratic equation on a Cartesian plane to visually identify the x-intercepts, offering a method to estimate the roots.

QHow does changing the 'c' coefficient affect the graph?

Changing the 'c' coefficient in the quadratic equation ax^2 + bx + c shifts the graph up or down without altering the shape or the direction of the parabola.

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Quadratic Equations Flashcards

Revise key terms and definitions from Quadratic Equations with interactive flashcards. Quick recall practice for CBSE Class 10 Mathematics.

Fast recall cards for definitions, methods, discriminant rules, and common patterns in quadratic equations.

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Standard form?

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ax^2 + bx + c = 0, where a ≠ 0.

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What are a, b, c?

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a: coefficient of x^2, b: coefficient of x, c: constant term.

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Reason for a ≠ 0

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If a=0, equation becomes linear (degree 1), not quadratic.

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Root of quadratic

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A value of x that makes ax^2+bx+c equal to 0.

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Roots equal zeros of?

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Zeros of p(x)=ax^2+bx+c are the roots of ax^2+bx+c=0.

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First step in solving

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Simplify and bring all terms to one side so RHS becomes 0.

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Why area gives quadratic?

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Area = length × breadth; product of two linear expressions expands to x^2 term.

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Choose variable wisely

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Let x represent the simplest unknown; express other quantities in terms of x and note constraints (e.g., x>0).

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If AB=0 then

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A=0 or B=0. Used after factorisation.

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Solve by factorisation

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Write quadratic as (px+q)(rx+s)=0 and solve each linear factor.

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For x^2+bx+c

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Find m,n with m+n=b and mn=c, then (x+m)(x+n)=0.

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For ax^2+bx+c

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Find two numbers with product a·c and sum b; split bx and factor by grouping.

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Factor by grouping

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After splitting: group pairs, factor common factors, then factor common binomial.

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After solving, do

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Substitute roots into original equation/situation to verify and reject invalid contextual roots.

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D equals

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D = b^2 - 4ac.

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If D>0

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Two distinct real roots; parabola cuts x-axis at two points.

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If D=0

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Real and equal roots; parabola touches x-axis once.

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If D<0

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No real roots; parabola does not meet x-axis.

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D is perfect square

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With integer coefficients, roots are rational because √D is rational.

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Formula for roots

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x = (−b ± √(b^2−4ac)) / (2a). Works for all quadratics.

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Does sign of b matter in b^2?

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No. b^2 is always non-negative; the sign disappears when squaring.

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If c is negative

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Then −4ac becomes positive (since a·c is negative), often increasing D.

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If c=0 then

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ax^2+bx= x(ax+b)=0 ⇒ one root is x=0.

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x^2 - k^2

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Factor as (x−k)(x+k).

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x^2 - 2kx + k^2

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(x−k)^2; equal roots at x=k.

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x^2 + 2kx + k^2

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(x+k)^2; equal roots at x=−k.

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D and x-intercepts

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D>0 two intercepts, D=0 one touching intercept, D<0 no intercepts.

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Most common sign slip

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Forgetting the minus in D=b^2−4ac or mishandling negative c.

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Fastest method?

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Factorise if easy; otherwise use formula. Use D for nature-of-roots questions.

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Word problems roots

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Keep only roots that satisfy conditions like positive length/time and the original statement.

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Need numbers with

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Product a·c and sum b (for ax^2+bx+c) to split bx.

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Quadratic roots count

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A quadratic can give two values; in applications one may be rejected by constraints.

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Quick verification

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Substitute x into LHS; if it becomes 0, the value is a root.

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D non-square but >0

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Roots are real and distinct but often irrational because √D is irrational.

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