All the ways to solve a quadratic — factorising, the formula, rearranging first, a leading coefficient, and quadratics hidden in fractions — plus a worked example for each and a full worksheet.
A quadratic equation has the form ax2 + bx + c = 0 (highest power 2). This guide covers every method and every exam variation. The order to try: (1) get one side = 0, (2) factorise if you can, (3) otherwise use the formula.
1Method 1: Factorising (the zero-product rule)
If a product equals zero, at least one factor must be zero. So factorise, set each bracket to zero, and solve.
- 1Factorise: two numbers that multiply to 6, add to −5 → −2 and −3 → (x − 2)(x − 3) = 0.Standard trinomial factorising.
- 2Set each bracket to zero: x − 2 = 0 or x − 3 = 0.Zero-product rule.
- 3x = 2 or x = 3.A quadratic usually has two solutions.
- 1Take out x: x(x − 7) = 0.Don't divide by x — you'd lose the root x = 0.
- 2x = 0 or x − 7 = 0.Zero-product rule.
- 3x = 0 or x = 7.Both are valid roots.
2Method 2: The quadratic formula
When it won't factorise, use the formula (on your data sheet). It works for every quadratic.
- 1a = 2, b = 3, c = −1.Match to ax2 + bx + c = 0.
- 2x = (−3 ± 9 + 8) / 4 = (−3 ± 17) / 4.b2 − 4ac = 9 − 4(2)(−1) = 17.
- 3x ≈ 0,28 or x ≈ −1,78.Work out both the + and − versions; round to 2 decimals.
3Type 3: Rearrange to standard form first
- 1Move everything to one side: x2 − 3x − 10 = 0.The equation must equal 0 before solving.
- 2Factorise: (x − 5)(x + 2) = 0.Multiply to −10, add to −3 → −5 and +2.
- 3x = 5 or x = −2.
4Type 4: A leading coefficient
- 1ac = 24; two numbers multiply to 24, add to −10 → −6 and −4.The ac-method for factorising when a ≠ 1.
- 2Split: 3x2 − 6x − 4x + 8 = 3x(x − 2) − 4(x − 2) = (x − 2)(3x − 4).Factorise by grouping.
- 3x = 2 or 3x − 4 = 0 → x = 43.
5Type 5: Quadratics hidden in fractions
- 1Multiply every term by x: x2 + 6 = 5x.Clear the fraction (note x ≠ 0).
- 2Rearrange: x2 − 5x + 6 = 0.Standard form.
- 3(x − 2)(x − 3) = 0 → x = 2 or x = 3.Both are non-zero, so both are valid.
Always make one side zero first, then try factorising before the formula. Never divide both sides by x — you lose the root x = 0; take x out as a common factor instead.
Don't lose a solution. A quadratic normally has two answers — writing only one is a common way to drop marks. And check any restrictions (like x ≠ 0) when you multiplied out a fraction.
Quadratics run through functions, calculus and geometry, so fluency here pays off everywhere. Work every type in the worksheet.
Practice exercises
Work each one out, then click to reveal the answer.
- 1Solve x2 − 7x + 12 = 0. (factorise)
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(x−3)(x−4)=0 → x = 3 or 4 - 2Solve x2 + 2x − 8 = 0. (factorise)
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(x+4)(x−2)=0 → x = −4 or 2 - 3Solve x2 − 9x = 0. (common factor)
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x(x−9)=0 → x = 0 or 9 - 4Solve x2 − 16 = 0. (difference of squares)
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(x−4)(x+4)=0 → x = ±4 - 5Solve x2 + 4x + 1 = 0 (formula, 2 dp). (formula)
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x = (−4±12)/2 ≈ −0,27 or −3,73 - 6Solve 2x2 − 5x + 1 = 0 (formula, 2 dp). (formula)
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x = (5±17)/4 ≈ 2,28 or 0,22 - 7Solve x2 = 2x + 15. (rearrange)
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x2−2x−15=0 → (x−5)(x+3)=0 → x = 5 or −3 - 8Solve 2x2 + 7x + 3 = 0. (leading coeff)
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(2x+1)(x+3)=0 → x = −12 or −3 - 9Solve x + 5x = 6. (fraction)
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x2−6x+5=0 → (x−1)(x−5)=0 → x = 1 or 5 - 10Solve (x − 2)(x + 1) = 4. (expand first)
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x2−x−2=4 → x2−x−6=0 → (x−3)(x+2)=0 → x = 3 or −2
Now practise it
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