Solving Quadratic Equations
A quadratic equation, like x² − 5x + 6 = 0, usually has two answers. This page gives you three ways to find them: factorising, completing the square and the quadratic formula. It shows why each one works, what the answers mean on a graph, and how to choose the best method for the question in front of you.
GCSE subject content: solve quadratic equations algebraically by factorising; find approximate solutions using a graph; Higher: solve by completing the square and by using the quadratic formula; identify turning points by completing the squareBefore you start. You need to be able to factorise x² + bx + c (see Double brackets and factorising quadratics) and to know what the graph of a quadratic looks like: a symmetrical U-shaped curve called a parabola (see Quadratic graphs). Remember that every positive number has two square roots: √9 = 3, but both 3² and (−3)² make 9.
1 · What "solve" means: roots on a graph
A quadratic equation has an x² term and no higher power, and you want to find the value(s) of x that make it true. To solve ax² + bx + c = 0 means to find every x that makes the expression equal zero. On the graph of y = ax² + bx + c, those are the points where y = 0: where the curve crosses the x-axis. The solutions are also called the roots.
Drag the plum point along the curve. Where does y become 0? Then change the sliders: a parabola can cross the x-axis twice, touch it once, or miss it completely. That's why a quadratic has two, one or no solutions.
Reading roots off a graph only gives approximate answers, often to 1 decimal place. The algebraic methods below give exact answers.
2 · Solving by factorising
If two numbers multiply to make 0, at least one of them must be 0. This is the zero-product rule. So once a quadratic is factorised, (x − 2)(x − 3) = 0, either x − 2 = 0 or x − 3 = 0, giving x = 2 or x = 3. You met this in Solving quadratics by factorising; here's a quick step-through, including the traps.
Two traps. (1) The equation must equal zero before you factorise: x(x + 2) = 15 does not mean x = 15 or x + 2 = 15. (2) Never divide both sides by x: from x² = 7x you'd get x = 7 and lose the solution x = 0.
3 · Completing the square Higher
Most quadratics don't factorise with whole numbers, so we need another method. Completing the square rewrites x² + bx + c in the form (x + p)² + q. Here's the picture behind it. x² is a square tile; bx is a strip of width b. Cut the strip in half, move one half underneath, and you almost have a bigger square, of side x + b/2. One small corner is missing: a square of side b/2.
So adding the corner completes the square, and then we must take it away again to keep the value the same:
x² + bx = (x + b/2)² − (b/2)²
Recipe: halve b to get the number in the bracket, then subtract that number squared. Any constant c just comes along for the ride: x² + 6x + 5 = (x + 3)² − 9 + 5 = (x + 3)² − 4.
4 · Turning points and solving from the completed square Higher
The completed form tells you the turning point (the lowest point of a U-shaped parabola, also called the vertex). In y = (x + p)² + q, the squared bracket is never negative, so it is smallest (zero) when x = −p. Then y = q. So the turning point is (−p, q). Notice the sign flips for the x-coordinate but not for the y-coordinate.
Try it. Write the expression in the form (x + p)² + q.
To solve by completing the square, rearrange to (x + p)² = number, then square-root both sides. Remember the ±: both the positive and negative square roots work.
5 · The quadratic formula Higher
Complete the square on the general equation ax² + bx + c = 0 once, with letters, and you get a formula that solves every quadratic. Step through the derivation; each line uses only the completing-the-square recipe from section 3.
x = −b ± √(b² − 4ac)2a
Learn it by heart: it solves every quadratic, including ones that don't factorise. Use it like this: write down a, b and c with their signs, substitute with brackets around negatives, work out the part under the square root first, then do the + version and the − version.
Formula mistakes examiners see every year
- Negative b: if b = −3 then −b = +3, and b² = (−3)² = +9, not −9. Type (−3)² with brackets on a calculator.
- The fraction line goes under everything: divide the whole of −b ± √(…) by 2a, not just the square root.
- 2a, not a: the bottom is 2 × a.
- Rounding too early: keep the full square root in your calculator until the last line.
Practice. Solve with the formula. Give both answers to 2 decimal places (either order).
6 · The discriminant: how many roots? Higher
The part under the square root, b² − 4ac, is called the discriminant. It "discriminates" between the three cases before you do any more work:
- b² − 4ac > 0: two different square roots (±), so two roots. The curve crosses the x-axis twice.
- b² − 4ac = 0: √0 = 0, so + and − give the same answer: one (repeated) root. The curve just touches the x-axis at its turning point.
- b² − 4ac < 0: you can't square-root a negative number, so there are no real roots. The curve never reaches the x-axis.
7 · Choosing a method and giving your answer
| If the equation… | use | example |
|---|---|---|
| has no x term, or is already (x + p)² = k | square-root both sides (±) | x² = 20 → x = ±2√5 |
| factorises easily (product/sum pair exists) | factorising | x² − x − 12 = 0 → x = 4, −3 |
| says "complete the square" or a = 1 with even b, exact answer wanted Higher | completing the square | x² + 8x + 3 = 0 → x = −4 ± √13 |
| says "to 2 d.p." or "3 s.f.", or won't factorise Higher | the quadratic formula | 2x² + x − 4 = 0 |
A question that asks for an answer "to 2 decimal places" is a hint that the quadratic will not factorise. "Give your answer in surd form" means leave the square root in, simplified: √20 = √(4 × 5) = 2√5, and cancel common factors: (6 ± 2√5)/2 = 3 ± √5.
Worked examples
Solve x² − 7x + 10 = 0
Solve x² + 6x + 4 = 0, giving your answers in surd form Higher
Find the turning point of y = x² − 8x + 3 Higher
Solve 2x² − 3x − 7 = 0, giving answers to 2 d.p. Higher
Solve 3x² + 5x − 1 = 0, leaving your answer in surd form Higher
Exam-style: a rectangle is x cm wide and (x + 3) cm long. Its area is 20 cm². Find x to 2 d.p. Higher
How many real roots does 2x² + 4x + 5 = 0 have? Higher
Quick check
What you've learned
- The solutions (roots) of ax² + bx + c = 0 are where y = ax² + bx + c crosses the x-axis. A quadratic has two, one or no real roots.
- Factorising: get = 0 first, factorise, then set each bracket to 0. Never divide by x.
- Higher Completing the square: x² + bx + c = (x + b/2)² − (b/2)² + c. The turning point of y = (x + p)² + q is (−p, q). Solve by isolating the square and taking ± square roots.
- Higher The quadratic formula x = (−b ± √(b² − 4ac)) / 2a comes from completing the square on ax² + bx + c = 0. Brackets around negatives; divide the whole top by 2a.
- Higher The discriminant b² − 4ac is positive (2 roots), zero (1 root) or negative (no real roots).
- "2 d.p." suggests the formula; "surd form" means leave simplified square roots; in context, reject answers that make no sense (negative lengths).