Completing the Square
Teach Genie
Review Completing the Square by teaching Genie
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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?
Completing the Square Revision Guide
Revision notes for CCEA GCSE Maths Completing the Square: explanations and worked examples.