Explore the world of quadratic equations, learning to solve them using various methods like factoring, completing the square, and the quadratic formula.
Quadratic Equations - Quick Look Revision Guide
Your 1-page summary of the most exam-relevant takeaways from Mathematics.
This compact guide covers 20 must-know concepts from Quadratic Equations aligned with Class X preparation for Mathematics. Ideal for last-minute revision or daily review.
Complete study summary
Essential formulas, key terms, and important concepts for quick reference and revision.
Key Points
Define quadratic equation.
A quadratic equation in variable x is of the form ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0. Example: 2x² + x - 3 = 0.
Standard form of quadratic equation.
The standard form is ax² + bx + c = 0, where a, b, c are coefficients. It's essential for solving and analyzing quadratic equations.
Roots of quadratic equation.
Roots are the values of x that satisfy the equation ax² + bx + c = 0. They can be real or complex numbers.
Factorisation method to find roots.
Split the middle term and factorise to find roots. Example: x² - 5x + 6 = 0 factors to (x-2)(x-3)=0, giving roots x=2, x=3.
Quadratic formula for roots.
Roots can be found using x = [-b ± √(b² - 4ac)] / 2a. This formula works for any quadratic equation.
Discriminant determines nature of roots.
Discriminant D = b² - 4ac. If D>0, two distinct real roots; D=0, two equal real roots; D<0, no real roots.
Sum and product of roots.
For ax² + bx + c = 0, sum of roots = -b/a, product of roots = c/a. Useful for finding equations from roots.
Real-world application: Area problems.
Quadratic equations model area problems, like finding dimensions of a rectangle given area and perimeter relations.
Real-world application: Projectile motion.
The path of projectiles can be modeled by quadratic equations, helping calculate maximum height or distance.
Completing the square method.
Transform the equation into (x+p)² = q form to solve. Useful when factorisation is not straightforward.
Nature of roots: Real and equal.
When discriminant D=0, the equation has exactly one real root (or two identical roots).
Nature of roots: No real roots.
If D<0, roots are complex and not real. The graph does not intersect the x-axis.
Graph of quadratic equation.
The graph is a parabola. If a>0, it opens upwards; if a<0, it opens downwards. Vertex is the turning point.
Vertex form of quadratic equation.
y = a(x-h)² + k, where (h,k) is the vertex. Useful for graphing and analyzing the parabola's properties.
Maximum and minimum values.
For a quadratic equation, the vertex gives the maximum (if a<0) or minimum (if a>0) value of the function.
Solving word problems.
Translate the problem into a quadratic equation by defining variables and setting up relationships based on given conditions.
Difference of distances problem.
Problems involving the difference of distances from two points can lead to quadratic equations, like in the prayer hall example.
Consecutive integers problems.
Finding consecutive integers whose product or sum of squares is given can be solved using quadratic equations.
Age-related problems.
Quadratic equations can model situations involving ages, especially when dealing with products or differences over time.
Checking solutions.
Always verify roots by substituting back into the original equation to ensure they satisfy it, especially in word problems.
Real Numbers encompass all rational and irrational numbers, forming a complete and continuous number line essential for various mathematical concepts.
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