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Completing the square means rewriting a quadratic so that it contains a squared bracket. Instead of a form like x2+px+qx^2+px+qx2+px+q, we aim for something like a(x+h)2+ka(x+h)^2+ka(x+h)2+k, which is easier to read from a graph.
The logic is based on the expansion of a perfect square:
(x+r)2=x2+2rx+r2 (x+r)^2=x^2+2rx+r^2 (x+r)2=x2+2rx+r2Running this backwards shows that the constant inside the bracket is exactly half the coefficient of the xxx term.
To complete the square for x2+px+qx^2+px+qx2+px+q, you halve the coefficient ppp and subtract the square of that value to keep the expression balanced. This results in the standard form:
x2+px+q=(x+p2)2+q−(p2)2 x^2+px+q=(x+\frac{p}{2})^2+q-\left(\frac{p}{2}\right)^2 x2+px+q=(x+2p)2+q−(2p)2Question 1
3 marksWrite x2−6x+1x^2 - 6x + 1x2−6x+1 in the form (x+a)2+b(x + a)^2 + b(x+a)2+b where a a\,a and b b\,b are integers.
In the expansion (x+r)2=x2+2rx+r2(x+r)^2 = x^2 + 2rx + r^2(x+r)2=x2+2rx+r2, how is the number inside the bracket (rrr) related to the xxx coefficient?
Revision notes for Edexcel GCSE Maths Completing the Square. Open the guide for explanations and worked examples. Written against the Edexcel GCSE Maths (1MA1) specification, so the content matches what's examinable rather than general Maths background.
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Rewrite x2+6x+2x^2+6x+2x2+6x+2 in completed square form.
