Solving quadratic equations: three methods
A quadratic equation has the form ax² + bx + c = 0. There are three reliable methods for solving it; you choose based on what's quickest.
Method 1: Factorising
If the quadratic factorises into two brackets, set each bracket equal to zero (the null factor law: if a product equals zero, at least one factor must be zero).
x² + 7x + 12 = 0. Factor: (x + 3)(x + 4) = 0. So x = -3 or x = -4.
2x² + x - 6 = 0. Factor (split-and-group or trial): (2x - 3)(x + 2) = 0. So x = 3/2 or x = -2.
Method 2: The quadratic formula
For ax² + bx + c = 0:
x = (-b ± √(b² - 4ac))/(2a).
This always works — even when factorising fails.
Example: x² - 4x - 1 = 0. a = 1, b = -4, c = -1.
Discriminant: (-4)² - 4(1)(-1) = 16 + 4 = 20.
x = (4 ± √20)/2 = (4 ± 2√5)/2 = 2 ± √5.
So x = 2 + √5 ≈ 4.24 or x = 2 - √5 ≈ -0.24 (3 s.f.).
Method 3: Completing the square
Useful for exact (surd) solutions and to find vertex form. From the completed square (x + p)² = q, take square roots: x + p = ±√q, so x = -p ± √q.
x² + 6x + 4 = 0. Complete: (x + 3)² - 9 + 4 = 0 ⇒ (x + 3)² = 5 ⇒ x = -3 ± √5.
The discriminant tells you what to expect
Δ = b² - 4ac:
Δ > 0: two distinct real roots.Δ = 0: one repeated real root.Δ < 0: no real roots (curve doesn't cross the x-axis).
✦Worked example— Worked example — choosing a method
Solve x² - 5x + 6 = 0.
- Try factorising first: 2 and 3 multiply to 6, sum to 5.
(x - 2)(x - 3) = 0.x = 2orx = 3. (Don't reach for the formula when factoring works in 5 seconds.)
Solve 2x² - 3x - 7 = 0.
- Doesn't factor nicely. Use the formula: a = 2, b = -3, c = -7.
- Discriminant: 9 + 56 = 65.
x = (3 ± √65)/4≈ (3 ± 8.062)/4 → ≈ 2.77 or ≈ -1.27.
⚠Common mistakes— Common mistakes (examiner traps)
- Forgetting the ± when square-rooting.
x² = 25 ⇒ x = ±5, not justx = 5. - Dropping a solution. Quadratics typically have two solutions; report both.
- Sign error in the discriminant.
-4acincludes the sign of c. With c negative,-4acis positive. - Bracket error in formula.
b²— careful with negative b:(-4)² = 16, not -16. - Premature rounding. Round only at the end; keep surds until the answer.
➜Try this— Quick check
Solve (x + 2)² = 9. Take root: x + 2 = ±3. So x = 1 or x = -5.
AI-generated · claude-opus-4-7 · v3-deep-algebra