Worked example

Rewrite \(y = {x^2} - 6x + 11\) in the form \(y = {(x - b)^2} + c\).

To get \(b\) (the number inside the bracket), halve the coefficient (number in front) of the second term in the original equation.

Half of -6 is -3 so the bracket becomes \((x-3)\).

\(y = {(x - 3)^2} - 9 + 11\)

\(y = {(x - 3)^2} + 2\)

Look again at the working above.

\(x^{2}-6x\) is not the same as \((x-3)^{2}\).

Expanding \({(x - 3)^2}\), we get:

\(= (x - 3)(x - 3)\)

\(= {x^2} - 3x - 3x + 9\)

\(= {x^2} - 6x + 9\)

So the \({x^2} - 6x\) matches the first two terms from the original equation and we have also added on an extra \(9\) from this squared bracket.

This needs to be taken off again to ensure that the value of the expression is not changed.

We then combine it with the +11 to get the final answer \(y = {(x - 3)^2} + 2\).

Question

Rewrite \(y = {x^2} + 10x + 7\) in the form \(y = {(x - b)^2} + c\).

Rewrite \(y=x^{2}+8x-3\) in the form of \(y=(x-b)^{2}+c\).