Wize High School Grade 11 Math Textbook > Completing the Square

Exploring Perfect Squares
We call a number a perfect square if it is the square of a number.

Examples
111 is a perfect square because 1=121=1^21=12
444 is a perfect square because 4=224=2^24=22
999 is a perfect square because 9=329=3^29=32...

PAGE BREAK
How about for Algebraic Expressions?
We call an algebraic expression a perfect square if it can be written as the square of another expression.

Example
(x+1)2\left(x+1\right)^2(x+1)2 is a perfect square. This expands and simplifies to
(x+1)(x+1)=x2+x+x+1=x2+2x+1\begin{array}{ll}
&\left(x+1\right)\left(x+1\right)\\[0.5em]
=&x^2+x+x+1\\[0.5em]
=&x^2+2x+1
\end{array}==​(x+1)(x+1)x2+x+x+1x2+2x+1​

PAGE BREAK
Creating Perfect Squares
How do we turn x2+bxx^2+bxx2+bx into a perfect square?
1. Let's represent this using algebra tiles


2. Cut the bxbxbx in half

3. Rearrange your tiles so you get a square with side lengths x+b2x+\dfrac{b}{2}x+2b​:

4. FIll in the missing piece with (b2)2\left(\dfrac{b}{2}\right)^2(2b​)2
So, we get the perfect square (x+b2)2=x2+bx+(b2)2\boxed{\left(x+\dfrac{b}{2}\right)^2=x^2+bx+\left(\dfrac{b}{2}\right)^2}(x+2b​)2=x2+bx+(2b​)2​

Write it Down
To turn x2+bxx^2+bxx2+bx into a perfect square, 
we add (b2)2\left(\dfrac{b}{2}\right)^2(2b​)2 to get x2+bx+(b2)2x^2+bx+\left(\dfrac{b}{2}\right)^2x2+bx+(2b​)2.
Then we can rewrite it as (x+b2)2\left(x+\dfrac{b}{2}\right)^2(x+2b​)2

Practice: Exploring Perfect Squares
Each of the following expressiosn are written in the form y=x2+bxy=x^2+bxy=x2+bx.

a) Identify the value of bbb.

b) What is the value of b2\dfrac{b}{2}2b​?

c) What is the value of (b2)2\left(\dfrac{b}{2}\right)^2(2b​)2?

x2+8xx^2+8xx2+8x

b=

\dfrac{b}{2}=

\left(\dfrac{b}{2}\right)^2=

I don't know

0:00 / 0:00

Example: Perfect Squares
For each of the following expressions, add a constant to make it into a perfect square. Then, factor this perfect square.

a) x2+4xx^2+4xx2+4x

The bbb value here is 444, so we can add (b2)2\left(\dfrac{b}{2}\right)^2(2b​)2 which is (42)2\left(\dfrac{4}{2}\right)^2(24​)2 to make this a perfect square x2+4x+(42)2x^2+4x+\left(\dfrac{4}{2}\right)^2x2+4x+(24​)2.

We can then rewrite this as the perfect square (x+42)2\left(x+\dfrac{4}{2}\right)^2(x+24​)2, which simplifies to (x+2)2\boxed{(x+2)^2}(x+2)2​.

b) x2−6xx^2-6xx2−6x

The bbb value here is −6-6−6, so we can add (b2)2\left(\dfrac{b}{2}\right)^2(2b​)2 which is (−62)2\left(\dfrac{-6}{2}\right)^2(2−6​)2 to make this a perfect square x2−6x+(−62)2x^2-6x+\left(\dfrac{-6}{2}\right)^2x2−6x+(2−6​)2.

We can then rewrite this as the perfect square (x+−62)2\left(x+\dfrac{-6}{2}\right)^2(x+2−6​)2, which simplifies to (x−3)2\boxed{(x-3)^2}(x−3)2​.

c) x2+5xx^2+5xx2+5x

The bbb value here is 555, so we can add (b2)2\left(\dfrac{b}{2}\right)^2(2b​)2 which is (52)2\left(\dfrac{5}{2}\right)^2(25​)2 to make this a perfect square x2+5x+(52)2x^2+5x+\left(\dfrac{5}{2}\right)^2x2+5x+(25​)2.

We can then rewrite this as the perfect square (x+52)2\boxed{\left(x+\dfrac{5}{2}\right)^2}(x+25​)2​.

Practice: Creating Perfect Squares
For x2−10xx^2-10xx2−10x,

a) determine what constant needs to be added to make it a perfect square.

b) Factor this perfect square.

a) What constant needs to be added to make this a perfect square?

b) Factor this perfect square.

I don't know

Practice: Creating Perfect Squares
For x2+7xx^2+7xx2+7x

a) determine what constant needs to be added to make it a perfect square.

b) Factor this perfect square.

a) What constant needs to be added to make this a perfect square? (Enter it as a decimal if needed)

b) Factor this perfect square.

I don't know