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Formula Sheet
Explore the world of quadratic equations, learning to solve them using various methods like factoring, completing the square, and the quadratic formula.
Quadratic Equations – Formula & Equation Sheet
Essential formulas and equations from Mathematics, tailored for Class X in Mathematics.
This one-pager compiles key formulas and equations from the Quadratic Equations chapter of Mathematics. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Formulas
Standard form of Quadratic Equation: ax² + bx + c = 0
a, b, c are real numbers with a ≠ 0. x represents the variable. This form is used to represent quadratic equations for solving and analysis.
Discriminant: D = b² - 4ac
D determines the nature of roots. If D > 0, two distinct real roots; D = 0, two equal real roots; D < 0, no real roots.
Quadratic Formula: x = [-b ± √(b² - 4ac)] / (2a)
Provides roots of the quadratic equation. ± indicates two possible solutions. Essential for solving equations not factorable easily.
Sum of roots: α + β = -b/a
α and β are the roots. This relation helps in finding roots quickly without solving the entire equation.
Product of roots: αβ = c/a
Another relation between roots and coefficients. Useful for constructing equations when roots are known.
Nature of roots based on Discriminant
D > 0: Real and distinct; D = 0: Real and equal; D < 0: No real roots. Crucial for understanding equation solutions without solving.
Factorization method for solving quadratic equations
Express the equation as (px + q)(rx + s) = 0. Solutions are x = -q/p and x = -s/r. Best for equations easily factorable.
Completing the square method
Transform equation into (x + p)² = q form. Solving gives x = -p ± √q. Useful when factorization is complex.
Graphical representation of quadratic equations
Parabola shape. Vertex form y = a(x - h)² + k helps in plotting. Vertex at (h, k). Aids in visualizing solutions.
Relation between coefficients and roots
For ax² + bx + c = 0, sum of roots is -b/a, product is c/a. Helps in quick verification of roots.
Equations
Example equation: x² - 5x + 6 = 0
Solutions are x = 2 and x = 3. Found by factorization: (x-2)(x-3)=0. Demonstrates basic solving technique.
Example equation: 2x² + x - 300 = 0
Represents a real-world problem like area calculation. Solved using quadratic formula or factorization.
Example equation: x² + 3x + 1 = 0
Discriminant D = 5 > 0, so two real roots. Illustrates use of discriminant to predict nature of roots.
Example equation: x² - 45x + 324 = 0
Represents a problem of finding two numbers. Roots give the numbers. Shows application in number problems.
Example equation: 6x² - x - 2 = 0
Solved by factorization to find x = 2/3 and x = -1/2. Demonstrates solving quadratic equations.
Example equation: x² + 7x - 60 = 0
Real-world scenario like distance calculation. Roots are x = 5 and x = -12. Negative root is ignored in context.
Example equation: 3x² - 2x + 1/3 = 0
Discriminant D = 0, indicating equal roots. Roots are x = 1/3. Shows case of equal roots.
Example equation: 2x² - 6x + 3 = 0
D = 12 > 0, two distinct real roots. Solved using quadratic formula. Illustrates discriminant's role.
Example equation: kx(x - 2) + 6 = 0
For equal roots, discriminant must be zero. Leads to finding k. Demonstrates condition for equal roots.
Example equation: x² - 55x + 750 = 0
Represents cost and production problem. Roots give production quantities. Application in economics.
Real Numbers encompass all rational and irrational numbers, forming a complete and continuous number line essential for various mathematical concepts.
Explore the world of Polynomials, understanding their types, degrees, and operations to solve algebraic expressions and equations effectively.
Explore the methods to solve a pair of linear equations in two variables, including graphical, substitution, elimination, and cross-multiplication techniques.
A chapter that explores sequences where each term after the first is obtained by adding a constant difference, focusing on their properties, nth term, and sum formulas.
