Completing the Square in Math
Examples & Formula for completing the square

What does "completing the square" do for us?

Answer
It gives us a way to find the last term of a perfect square trinomial .

Okay, what is a "perfect square trinomial"??

Answer
A perfect square trinomial is a polynomial that you get by squaring a binomial.(binomials are things like 'x + 3' or 'x − 5')
Examples of perfect square trinomials
 $$ x^2 + 2x + 1$$ This is a perfect square trinomial because it equals $$(x+ 1)^2$$
 $$x^2 + 4x + 4$$ This is a perfect square trinomial because it equals $$(x+ 2)^2$$
 $$x ^2 + 6x + 9 $$ This is a perfect square trinomial because it equals $$(x+ 3)^2$$
Non Examples
These are NOT perfect square trinomials
 $$x ^2 + 3x + 2 $$ This is not a perfect square trinomial because it equals $$ (x+ 1)(x + 2) $$
 $$x ^2 + 7x + 10 $$ This is not a perfect square trinomial because it equals $$ (x+ 2)(x + 5) $$
 $$ x ^2  9$$ This is not a perfect square trinomial because it equals $$ (x \color{Red}{+} 3) (x \color{Red}{} 3 ) $$

So, what does it really do for us?

Answer
Before we look answer that, let's first examine the three equations below. How would you solve each one?
Formula for Completing the Square
To best understand the formula and logic behind completing the square, look at each example below and you should see the pattern that occurs whenever you square a binomial to produce a perfect square trinomial.
Practice Problems
Directions Find the missing value to complete the square
Problem 1) $$x^2 + \color{red}{16}x + \blacksquare $$
Step 1)
Divide the middle term by 2

$$
\\
\frac{ \color{red}{16} }{ \color{green}{2} }= \color{blue}{8}
$$

Step 2)
Then square the result

$$
\color{blue}{8}^2 = 64
$$

Final Answer! : $$x^2 + \color{red}{16}x + 64 $$
You can always check your work by seeing by foiling the answer to step 2 and seeing if you get the correct result
$$ (x+ \color{blue}{8}) =x^2 + \color{red}{16}x + 64 $$

II. Completing Square when a is not 1

What about $$\color{blue}{3}x^2 + 18x$$, what then ?

Practice Problems
The next problems are quite challenging, good luck!
