Completing the Square
Completing the square is a technique for rewriting a quadratic expression x^2 + bx + c in the form (x + a)^2 + d, which makes it easy to find the turning point of a quadratic graph, solve a quadratic equation exactly, or prove an expression is always positive. It is tested throughout GCSE Higher algebra and graph questions.
Before you start
Make sure you're comfortable with these topics first:
Method
- Halve the coefficient of x to find a, so that x^2 + bx becomes (x + b/2)^2 minus (b/2)^2.
- Write the expression as (x + b/2)^2 - (b/2)^2 + c, then simplify the constant term.
- If the coefficient of x^2 is not 1, factorise it out of the x^2 and x terms first, complete the square inside the bracket, then multiply back out carefully.
- To solve an equation, set the completed square form equal to zero, isolate the squared bracket, and take the square root of both sides (remembering plus or minus).
- To find a turning point, read the coordinates directly from the completed square form: it occurs at x = -a, with value d.
- To prove an expression is always positive, show the squared bracket can never be negative, so adding a positive constant keeps the whole expression positive.
Worked example
Write x^2 + 10x + 18 in the form (x + a)^2 + b, and hence solve x^2 + 10x + 18 = 0, giving your answers as exact surds.
- Halve the coefficient of x: 10 / 2 = 5, so start with (x + 5)^2.
- Expand (x + 5)^2 = x^2 + 10x + 25, which is 25 - 18 = 7 more than the original constant term.
- Subtract this excess: x^2 + 10x + 18 = (x + 5)^2 - 7.
- Set the completed square equal to zero: (x + 5)^2 - 7 = 0, so (x + 5)^2 = 7.
- Take the square root of both sides: x + 5 = sqrt(7) or x + 5 = -sqrt(7).
- Final answer: x = -5 + sqrt(7) or x = -5 - sqrt(7).
Practice questions
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Exam-style questions
Written in the style of a GCSE Maths exam paper, with a full mark scheme.
Write x^2 + 14x + 24 in the form (x + a)^2 + b.
The height, h metres, of a firework t seconds after launch is modelled by h = -4t^2 + 24t + 2. (a) Write -4t^2 + 24t + 2 in the form -4(t - p)^2 + q. (b) Hence find the maximum height of the firework and the time at which it occurs.
Solve 2x^2 - 8x - 3 = 0 by completing the square. Give your answers in the form (a + sqrt(b)) / c, where a, b and c are integers.
Free printable worksheet
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