Quadratic Equations

Overview

Quadratic equations form a cornerstone of algebra and appear consistently in OTET Paper II 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 problems, profit-loss scenarios, and geometric relationships.

For OTET, you must master three core skills: recognising the standard form, finding roots using multiple methods, and applying the relationship between roots and coefficients. Questions typically test your ability to solve equations quickly, interpret the discriminant, and frame word problems as quadratic equations. This topic connects directly to algebra, mensuration, and even science applications involving motion and energy.

The emphasis is on conceptual clarity and computational accuracy. Expect 2–4 questions from this topic, often combined with real-life application scenarios relevant to upper primary teaching contexts.

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 a linear equation.

• **Roots/Solutions**: 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. This single value tells you everything about the roots without actually solving.

• **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 (roots are complex/imaginary)

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

• **Methods of Solving**: Factorisation, completing the square, and quadratic formula are the three standard methods. For exams, factorisation is fastest when applicable; the formula is reliable for all cases.

• **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 of quadratic equation | | x = (−b ± √(b² − 4ac)) / 2a | Quadratic formula—universal method for finding roots | | 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 |