Completing the Square Practice Questions

Write $x^2 - 12x + 5$ in the form $(x - a)^2 - b$, where $a$ and $b$ are integers.

Hence write down the equation of the line of symmetry of the curve with equation $y = x^2 - 12x + 5$.

Write $x^2 + 4x - 9$ in the form $(x + a)^2 + b$, where $a$ and $b$ are integers.

Hence write down

(i) the minimum value of $x^2 + 4x - 9$,

(ii) the value of $x$ at which this minimum value occurs.

Write $2x^2 + 10x + 4$ in the form $a(x + b)^2 + c$, where $a$, $b$ and $c$ are constants.

Hence write down the coordinates of the turning point on the curve with equation $y = 2x^2 + 10x + 4$, and state whether it is a maximum or a minimum point.

Write $60 + 16x - 2x^2$ in the form $a(x + b)^2 + c$, where $a$, $b$ and $c$ are integers.

You must show all your working.

Use your answer to part (a) to find the maximum weekly profit, and the number of pounds of cheese that must be sold to achieve it.

Show that $x^2 - 10x + 29$ can be written in the form $(x - a)^2 + b$, where $a$ and $b$ are positive integers.

Hence prove that $x^2 - 10x + 29$ is positive for all values of $x$.

Find, in terms of $k$, the coordinates of the turning point on the curve.

You must show all your working.

The minimum value of $y$ is $-61$.

Find the value of $k$.