Quadratic Equations

Overview

Quadratic equations form a foundational topic in algebra that appears consistently in Assam TET Paper II. A quadratic equation is a polynomial equation of degree 2, meaning the highest power of the variable is 2. Understanding how to find roots and analyse their nature is essential for solving problems quickly in the exam.

This topic connects directly to other areas like coordinate geometry (parabolas), physics (projectile motion), and practical problems involving area and optimization. For TET, you must be comfortable with the standard form, factorization methods, the quadratic formula, and the discriminant. Questions typically test your ability to find roots, determine their nature, and apply relationships between roots and coefficients.

Mastery of quadratic equations demonstrates algebraic reasoning—a key competency for upper primary mathematics teachers who must explain these concepts clearly to students.

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 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 or complex).

• **Discriminant (D)**: The expression D = b² − 4ac determines the nature of roots without actually solving the equation. This is a powerful shortcut for MCQs.

• **Nature of roots based on D**:
• D > 0 → Two distinct real roots
• D = 0 → Two equal real roots (one repeated root)
• D < 0 → No real roots (complex/imaginary roots)

• **Sum and product of roots**: If α and β are roots of ax² + bx + c = 0, then:
• Sum: α + β = −b/a
• Product: αβ = c/a

• **Methods to solve**: Factorization, completing the square, and quadratic formula. For exams, factorization is fastest when applicable; formula is the universal backup.

• **Forming equations from roots**: If roots are given as α and β, the equation is x² − (α + β)x + αβ = 0.

Formulas / Key Facts

| Formula | Context | |---------|---------| | ax² + bx + c = 0 | Standard form (a ≠ 0) | | x = (−b ± √(b² − 4ac)) / 2a | Quadratic formula—universal method | | D = b² − 4ac | Discriminant—determines nature of roots | | α + β = −b/a | Sum of roots | | αβ = c/a | Product of roots | | x² − (sum)x + (product) = 0 | Forming equation when roots are known |

**Quick facts to remember**: