# Completing the Square

Completing the square is when either of the following is performed:

• $x^2+ax+b$ is written in the form $(x+\alpha)^2+\beta$
• $ax^2+bx+c$ is written in the form $\alpha(x+\beta)^2+\gamma$

For the simpler case where the coefficient of $x^2$ is 1:

1. Take $\alpha$ to be half of $a$.
2. Expand $(x+\alpha)^2$.
3. Choose $\beta$ so as to adjust the constant so that the original quadratic expression is obtained.

See Example 1.

For the more complicated case where the coefficient of $x^2$ is not 1, remove a factor of $a$ from the original quadratic and perform the above on the inside of the brackets before expanding again in the final step. See Example 2.

The graph of a quadratic can easily be sketched if you think about the transformations that have been applied to the graph of $y=x^2$.

• Consider the graph of $y=x^2$
• The graph of say $y=(x+1)^2$ can be sketched by shifting the graph of $y=x^2$ to the left by 1. See x-transformations on the Transformations page.
• The graph of $y=3(x+1)^2$ can be sketched by stretching the graph of $y=(x+1)^2$ about the x-axis by a factor of 3. See y-transformations on the Transformations page.
• The graph of $y=3(x+1)^2-4$ can then be sketched by shifting the graph of $y=3(x+1)^2$ down by 4. See y-transformations again on the Transformations page.

# Completing the Square – Board Question

### Example 1

Write $x^2+4x+9$ in form $(x+\alpha)^2+\beta$.

### Example 2

Write $2x^2+8x-5$ in the form $p(x+q)^2+r$.

### Solomon Exercises

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