Solving quadratic equations
A quadratic equation has an x² term and usually two solutions. Factorise when you can; when you cannot, the quadratic formula always works. The solutions are where the graph crosses the x-axis.
Part 1 of 3: Learn it
In short
- Rearrange so one side is 0 before you solve.
- If (x − a)(x − b) = 0, then x = a or x = b.
- The quadratic formula always works when factorising does not.
Where this is in your specification
Spec points: DfE A4, A11 and A18 (the formula and completing the square are Higher tier)
| Board | Topic: Quadratics |
|---|---|
| DfE content | A4, A11, A12, A18, A22 |
| AQA 8300 | Algebra |
| Edexcel 1MA1 | Algebra |
| OCR J560 | Algebra |
Factorising x² + bx + c
Find two numbers that multiply to give c and add to give b. For x² + 7x + 12, the numbers are 3 and 4 (3 × 4 = 12, 3 + 4 = 7), so x² + 7x + 12 = (x + 3)(x + 4).
- Difference of two squares: x² − 64 = (x − 8)(x + 8).
- Common factor first: x² − 6x = x(x − 6).
- Check by expanding the brackets again.
Solving by factorising
If two things multiply to make zero, one of them must be zero. So once the equation is factorised, set each bracket equal to zero.
The quadratic formula (Higher)
For ax² + bx + c = 0:
Use it when a question asks for answers to a number of decimal places or significant figures: that is a sign the quadratic does not factorise. The part under the square root, b² − 4ac, tells you how many solutions there are: positive gives two, zero gives one, negative gives none.
Completing the square (Higher)
Halve the x coefficient and write the bracket, then take away its square. For example x² + 8x + 5 becomes (x + 4)² − 16 + 5, which is (x + 4)² − 11. This form also gives the turning point of the graph: here (−4, −11).
Factorise x² + x − 12.
Show the answer
(x + 4)(x − 3).
Part 2 of 3: See it worked
Worked examples
Example 1
Solve x² − 5x − 14 = 0.
- Two numbers that multiply to −14 and add to −5: −7 and +2
- (x − 7)(x + 2) = 0
- x − 7 = 0 or x + 2 = 0
Answer: x = 7 or x = −2.
Example 2
Solve 3x² − 2x − 4 = 0, giving your answers to 2 decimal places.
- a = 3, b = −2, c = −4
- Discriminant: (−2)² − 4 × 3 × (−4) = 4 + 48 = 52
- x = (2 ± √52) ÷ 6
- x = (2 + 7.2111) ÷ 6 = 1.5352 or x = (2 − 7.2111) ÷ 6 = −0.8685
Answer: x = 1.54 or x = −0.87.
Common mistakes
- Solving before the equation equals zero, for example x² + 3x = 10 without moving the 10.
- Getting the signs of the solutions the wrong way round: (x − 7) = 0 gives x = +7.
- Dividing only the square root part by 2a in the formula.
- Typing b² as −3² on a calculator when b = −3. Use brackets: (−3)² = 9.
Solve x² − 49 = 0.
Show the answer
x = 7 or x = −7.
Part 3 of 3: Test yourself
Check yourself
Answer each one in your head or on paper first, then open it to check.
Factorise x² + x − 12.
(x + 4)(x − 3).
Solve x² − 49 = 0.
x = 7 or x = −7.
How many solutions does x² + 2x + 5 = 0 have?
None: b² − 4ac = 4 − 20 = −16, which is negative.
Jobs that use this
- Actuary (se abre en otra pestaña)
- Software developer (se abre en otra pestaña)
- Architect (se abre en otra pestaña)
Each link opens the job profile on the National Careers Service (England). In the rest of the UK: My World of Work (Scotland), Careers Wales, nidirect careers (Northern Ireland).
Estos apuntes están en inglés porque siguen los programas de examen del Reino Unido.
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