Quadratic Equations

Overview

Quadratic equations form a cornerstone topic in the JKTET Paper II Mathematics section. These second-degree polynomial equations appear frequently in competitive exams because they test both algebraic manipulation skills and conceptual understanding of roots, discriminants, and the relationship between coefficients and solutions.

For JKTET, you must be comfortable with finding roots using factorisation, completing the square, and the quadratic formula. The discriminant concept is particularly exam-relevant as it determines the nature of roots without actually solving the equation. Questions typically involve direct solving, finding the nature of roots, or forming equations when roots are given.

Mastery of this topic also supports your understanding of related areas like algebraic expressions, coordinate geometry (parabolas), and word problems involving area, speed, or age calculations.

Key Concepts

• **Standard Form**: A quadratic equation is written as ax² + bx + c = 0, where a ≠ 0 and a, b, c are real numbers. The condition a ≠ 0 is essential—otherwise it becomes linear.

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

• **Discriminant (D)**: The expression D = b² − 4ac determines the nature of roots without solving. This is the single most tested concept in exams.

• **Sum and Product of Roots**: If α and β are roots, then α + β = −b/a and αβ = c/a. These relationships allow you to form equations or verify answers quickly.

• **Methods of Solving**: Three standard methods exist—factorisation (fastest when applicable), completing the square (conceptually important), and the quadratic formula (universal method).

• **Perfect Square Trinomial**: When D = 0, the quadratic is a perfect square, and both roots are equal (repeated root).

• **Symmetric Functions**: Expressions like α² + β², α³ + β³, or 1/α + 1/β can be calculated using sum and product without finding individual roots.

Formulas / Key Facts

**Quadratic Formula**: x = (−b ± √(b² − 4ac)) / 2a — Use when factorisation is not obvious.

**Discriminant**: D = b² − 4ac — Determines nature of roots.

**Nature of Roots based on D**:

• D > 0 → Two distinct real roots
• D = 0 → Two equal real roots (repeated root)
• D < 0 → No real roots (complex conjugate pair)

**Sum of Roots**: α + β = −b/a — Note the negative sign.