A quadratic equation has the form ax² + bx + c = 0, where a ≠ 0. There are three methods to solve them: factorisation (fastest when it works), completing the square, and the quadratic formula. Knowing all three lets you handle any question WAEC throws at you.
Method 1: Factorisation
Find two numbers that multiply to give ac and add to give b, then split the middle term. Example: Solve x² + 5x + 6 = 0 Find two numbers that multiply to 6 and add to 5: they are 2 and 3. So x² + 5x + 6 = (x + 2)(x + 3) = 0 Therefore x = -2 or x = -3
Method 2: Quadratic Formula
When factorisation fails (the equation doesn't factorise neatly), use: x = (-b ± √(b² - 4ac)) / 2a Example: Solve 2x² - 3x - 2 = 0 a = 2, b = -3, c = -2 x = (3 ± √(9 + 16)) / 4 = (3 ± √25) / 4 = (3 ± 5) / 4 x = (3+5)/4 = 2 OR x = (3-5)/4 = -0.5
The discriminant
The value b² - 4ac inside the square root is called the discriminant. It tells you about the nature of the roots: • If b² - 4ac > 0: two distinct real roots • If b² - 4ac = 0: one repeated root (the parabola just touches the x-axis) • If b² - 4ac < 0: no real roots
Key points
- Always rearrange to ax² + bx + c = 0 first
- Factorisation is fastest: try it first
- Quadratic formula works for all quadratics
- The discriminant tells you how many roots exist
- Check your answers by substituting back in
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WAEC tip
WAEC Paper 2 often asks for roots correct to 2 decimal places, which signals the quadratic formula. Memorise it: write it out ten times tonight if you need to.
Practise this
Solve 3x² + 7x - 6 = 0 using both factorisation and the formula. Check that both methods give the same answer.
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