Quadratic Equations

Overview

Quadratic equations form a cornerstone of algebra and appear frequently in MP TET Varg-2 Mathematics. A quadratic equation is a polynomial equation of degree 2, meaning the highest power of the variable is 2. These equations model numerous real-world situations—projectile motion, area calculations, profit-loss problems, and more.

For the MP TET exam, you must master identifying quadratic equations, finding their roots using multiple methods, understanding the nature of roots through the discriminant, and applying the relationship between roots and coefficients. Questions typically test computational accuracy and conceptual clarity, often mixing word problems with direct calculations.

This topic connects directly to algebraic expressions, factorisation, and linear equations. A solid grasp here also supports understanding of coordinate geometry and graphing parabolas at higher levels.

Key Concepts

• **Standard Form**: A quadratic equation in variable x is written as ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0. The condition a ≠ 0 is crucial—without it, the equation becomes linear.

• **Roots/Solutions**: The values of x that satisfy the equation are called roots or zeros. A quadratic equation has exactly two roots (which may be equal, real, or complex).

• **Discriminant (D)**: The expression D = b² − 4ac determines the nature of roots. It tells us whether roots are real and distinct, real and equal, or imaginary—without actually solving the equation.

• **Sum and Product of Roots**: If α and β are roots of ax² + bx + c = 0, then Sum (α + β) = −b/a and Product (αβ) = c/a. These relationships help form equations when roots are known.

• **Methods of Solution**: Four primary methods exist—factorisation, completing the square, quadratic formula, and graphical method. Factorisation is quickest when applicable; the quadratic formula works universally.

• **Parabola Connection**: The graph of y = ax² + bx + c is a parabola. The roots of the equation are the x-coordinates where the parabola crosses the x-axis.

Formulas / Key Facts

| Formula/Fact | Context | |--------------|---------| | ax² + bx + c = 0 (a ≠ 0) | Standard form of quadratic equation | | D = b² − 4ac | Discriminant formula | | D > 0 → Two distinct real roots | Parabola cuts x-axis at two points | | D = 0 → Two equal real roots | Parabola touches x-axis at one point | | D < 0 → No real roots (imaginary) | Parabola doesn't touch x-axis | | x = (−b ± √D) / 2a | Quadratic formula (Shreedharacharya's rule) | | Sum of roots = −b/a | α + β = −b/a | | Product of roots = c/a | αβ = c/a | | If roots are α, β, equation is x² − (α+β)x + αβ = 0 | Forming equation from roots |