The expression `(a + 3)^2` is an example of a perfect square and expands to `a^2 + 6a + 9`.

The reverse process - taking an expression and creating a perfect square with an additional constant - is called completing the square. This is useful as it gives the turning point of the graph of the function.

Take the expression `b^2 + 8b + 22`:

Halve the coefficient of the `b` term: 8 ÷ 2 = 4;

Create the squared term from the `b` and the 4: `(b + 4)`;

Expand the squared term: `(b + 4)^2 = b^2 + 8b + 16`;

Add an adjustment (6) to obtain the original expression:

`(b + 4)^2 + 6 = b^2 + 8b + 22`

The turning point of the quadratic graph is at (-4, 6): the `x`-coordinate is the value of b that makes the squared term zero; and the `y`-coordinate is the value of the adjustment.

If the squared term has a coefficient, square root the coefficient and use this inside the bracket. Then divide the `x` term by that same value; and halve as before:

`9x^2 + 30x + 13`

√9 = 3 to get the coefficient of the `x` term;

and 30 ÷ 3 = 10 and halved = 5;

Create the squared term: `(3x + 5)`;

Expand the squared term `(3x + 5)^2 = 9x^2 + 30x = 25`

Add a constant -12 for the original expression:

`(3x + 5)^2 - 12 = 9x^2 + 30x + 13`