Quadratic Equations

Overview

Quadratic equations form a cornerstone of upper-primary and secondary mathematics, appearing consistently in KAR TET Paper II. This topic tests both computational skills and conceptual understanding—you must be able to find roots using multiple methods, interpret the discriminant, and apply quadratic equations to word problems.

For TET preparation, focus on three areas: (1) solving equations using factorisation, completing the square, and the quadratic formula; (2) using the discriminant to determine the nature of roots without solving; and (3) understanding the relationship between roots and coefficients. Questions typically involve straightforward computation, but examiners also test whether candidates can identify which method suits a given problem and interpret results in real-world contexts.

Mastering quadratic equations also builds the foundation for coordinate geometry (parabolas), physics (projectile motion), and higher algebra—making this topic essential for both exam success and effective classroom teaching.

Key Concepts

• **Standard form**: A quadratic equation must be written as ax² + bx + c = 0, where a ≠ 0. The condition a ≠ 0 is crucial—if a = 0, the equation becomes linear.

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

• **Discriminant (D or Δ)**: The expression b² − 4ac determines the nature of roots without actually solving the equation.

• **Sum and product of roots**: If α and β are roots, then α + β = −b/a and αβ = c/a. This relationship helps verify answers and construct equations from given roots.

• **Methods of solving**: Factorisation (quickest when applicable), completing the square (foundational method), and quadratic formula (universal method).

• **Zero product property**: If the product of two factors equals zero, at least one factor must be zero. This underlies the factorisation method.

• **Graphical interpretation**: Roots represent x-intercepts of the parabola y = ax² + bx + c. The discriminant tells us whether the parabola crosses, touches, or misses the x-axis.

Formulas / Key Facts

**Quadratic Formula** x = (−b ± √(b² − 4ac)) / 2a Use when factorisation is difficult or when exact roots are needed.

**Discriminant** D = b² − 4ac

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

**Sum of Roots** α + β = −b/a