Solving quadratic equations three ways
A quadratic equation is any equation that can be written as ax² + bx + c = 0, where a is not zero. It might look intimidating, but there are just three tools for solving it, and you can usually tell which to reach for within a few seconds.
Method 1: Factorising
Best when the numbers are friendly. You are looking for two brackets that multiply out to the original.
Solve x² + 7x + 12 = 0. Find two numbers that multiply to 12 and add to 7: that is 3 and 4. So (x + 3)(x + 4) = 0. For a product to be zero, one bracket must be zero, so x = -3 or x = -4.
Method 2: The quadratic formula
Works every time, even when the numbers are ugly. The formula is x = (-b ± √(b² - 4ac)) ÷ 2a.
Solve 2x² - 3x - 5 = 0. Here a = 2, b = -3, c = -5. The discriminant is 9 + 40 = 49, and √49 = 7. So x = (3 ± 7) ÷ 4, which gives x = 2.5 or x = -1.
Method 3: Completing the square
Useful when you are asked for the vertex or need to rewrite the equation. Solve x² + 6x + 2 = 0.
- Move the constant: x² + 6x = -2.
- Add half the x-coefficient squared (3² = 9) to both sides: x² + 6x + 9 = 7.
- Write the left as a square: (x + 3)² = 7.
- Take square roots: x + 3 = ±√7, so x = -3 ± √7, about -0.35 or -5.65.
Which method should I use?
| Situation | Use |
|---|---|
| Small whole numbers, and b and c suggest an obvious pair | Factorising |
| Awkward numbers, decimals or a calculator allowed | Formula |
| You need the turning point or a rewritten form | Completing the square |
| Not sure | Formula, then check by substituting |
What the discriminant tells you
Before going any further, calculate b² - 4ac. If it is positive, expect two real answers. If it is zero, there is one repeated answer. If it is negative, the parabola never touches the x-axis and there are no real solutions. Knowing this in advance stops you wasting time on a factorisation that cannot work.
Always check
Put each answer back into the original equation. With x = 2.5 in 2x² - 3x - 5: 12.5 - 7.5 - 5 = 0. Correct. You can also confirm with the quadratic equation solver, which shows the discriminant, both roots and the vertex.
Check each of your answers with the quadratic solver.