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`

Complete the square for `4a^2 + 8a - 12`

Take the square root of the `x^2` coefficient: √4 = 2. This is the new `x` multiplier;

For the integer; divide by the new `x` multiplier 8 / 2 = 4; then halve 4 ÷ 2 = 2

Obtain `(2a + 2)^2 = 4a^2 + 8a + 4 `

Add a constant (-16) to obtain the original expression `(2a + 2)^2 - 16 = 4a^2 +8a - 12`

Answer: `(2a + 2)^2 - 16`

Complete the square for `b^2 - 10b`

For the integer: half the `b` coefficient, = -5

Therefore `(b - 5)^2 = b^2 - 10b + 25`

Add a constant -25 to obtain `(b - 5)^2 - 25 = b^2 - 10b`

Answer: `(b - 5)^2 - 25`

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