🔢 Quadratic Quest: Mastering Equations Efficiently!

Quadratic equations appear everywhere in O-Level Mathematics, from projectile motion to area problems. Being able to solve them quickly and accurately is **crucial** for securing marks in the Extended (E) paper. In this article, we’ll review the three powerhouse methods: factorising, the quadratic formula, and completing the square.

Cracking the Quadratic Code

A quadratic equation is any equation that can be written in the form

ax^2 + bx + c = 0

, where

a \neq 0

. The goal is to find the values of

x

(called the *roots*) that satisfy the equation. **Factorising** splits the quadratic into two linear factors when possible, the **quadratic formula** works for every quadratic, and **completing the square** rewrites the expression to reveal its roots or vertex.

Key point 1: Factorising is fastest when $b$ and $c$ are "nice" integers.
Key point 2: The quadratic formula $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ never fails, even when factorising is impossible.

Worked Example: Factor & Formula in Action

Problem: Solve

2x^2 - 5x - 3 = 0

using (i) factorising and (ii) the quadratic formula.

Step 1 (Factorising):
$2x^2 - 5x - 3 = 0 \;\Rightarrow\; (2x + 1)(x - 3) = 0$
Step 2 (Roots): $2x + 1 = 0 \Rightarrow x = -\dfrac{1}{2}$ , and $x - 3 = 0 \Rightarrow x = 3$

Master these techniques by practising a variety of quadratic equations. With consistent effort, you’ll turn every quadratic into quick marks on exam day!