Quadratic Equations Demystified: A Step-by-Step Guide for Grade 10–12 Students (CAPS & IEB)

Every year, quadratic equations trip up thousands of South African students — not because they're impossible, but because they're misunderstood. The good news: once the methods click, they become some of the most predictable marks in the paper. Here's how to get there.

Step 1: What is a quadratic equation?

In plain terms, it's an equation where the highest power is 2 — for example, ax² + bx + c = 0. A real-life version: working out the profit of a small spaza shop selling vetkoek at R5 each might give you something like −2x² + 50x − 100 = 0.

Two terms worth knowing:

• Coefficients: the numbers in front of x² and x, plus the constant term.
• Roots (solutions): the value or values of x that make the equation true.

Step 2: The three methods to solve any quadratic

Method 1: Factorising (the quick win)

When to use it: when the equation factors neatly, which is common in Grade 10. Take x² + 5x + 6 = 0.

1. Find two numbers that multiply to 6 and add to 5 → 2 and 3.
2. Write as factors: (x + 2)(x + 3) = 0.
3. Solve for x: x = −2 or x = −3.

A tip worth remembering: if it doesn't factor easily within about a minute, switch methods — don't waste exam time.

Method 2: The quadratic formula (the reliable backup)

When to use it: always works, and it's especially useful for the messier equations in Grade 11–12. The formula is x = (−b ± √(b² − 4ac)) / 2a.

Take 2x² − 4x − 6 = 0:

1. Identify a = 2, b = −4, c = −6.
2. Substitute into the formula.
3. Simplify to get x = 3 or x = −1.

Method 3: Completing the square

When to use it: when you're asked to explicitly, or for vertex-form problems. Take x² − 6x + 5 = 0:

1. Move the constant: x² − 6x = −5.
2. Take half of −6 → −3, square it → 9, add to both sides: x² − 6x + 9 = 4.
3. Factor the left side: (x − 3)² = 4.
4. Solve: x − 3 = ±2, so x = 5 or x = 1.

This method also helps with graphing parabolas — useful for Paper 1.

Step 3: Avoid these four costly mistakes

1. Forgetting the ± in the square root — it costs you half the marks.
2. Miswriting coefficients — double-check a, b and c.
3. Rushing factorisation — check that your factors expand back to the original equation.
4. Ignoring negative signs — remember that −b means the opposite sign of b.

Step 4: Practise with past papers

Try these, then check your working:

1. Factorise: x² + 9x + 20 = 0.
2. Quadratic formula: 3x² + 2x − 5 = 0.
3. Complete the square: x² + 8x + 7 = 0.

When to get help

If you find yourself asking "but why does the formula work?", that's exactly the kind of thing a tutor can unpack with visuals and real examples. No question is too basic. Bridge can match your child with a tutor to work through quadratics step by step — in-person or online — and the first assessment is free.