Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

Activity

1.

What is the factored form of $x^{2} + 6 x + 9$ ?

Compare your answer:

$(x + 3)^{2}$

2.

What is the factored form of $x^{2} - 10 x + 25$ ?

Compare your answer:

$(x - 5)^{2}$

3.

What is the standard form of $(x - 7)^{2}$ ?

Compare your answer:

$x^{2} - 14 x + 49$

4.

Enter the value of $c$ .

What is the value of $c$ in the expression $x^{2} - 20 x + c$ ?

100

5.

Enter the missing value of the factored form.

If $x^{2} - 20 x + c$ is factored to be $(x -)^{2}$ , what is the missing value of the factored form of this expression?

10

6.

Enter the value of $c$ .

What is the value of $c$ in the expression $x^{2} + 16 x + c$ ?

64

7.

Enter the missing value of the factored form.

If $x^{2} + 16 x + c$ is factored to be $(x -)^{2}$ , what is the missing value of the factored form of this expression?

8

8.

What is the value of $c$ in the expression $x^{2} + 7 x + c$ ?

Compare your answer:

$\frac{49}{4}$

9.

If $x^{2} + 7 x + c$ is factored to be $(x -)^{2}$ , what is the missing value of the factored form of this expression?

Compare your answer:

$\frac{7}{2}$

10.

What is the value of $c$ in the expression $x^{2} + b x + c$ ?

Compare your answer:

$(\frac{b}{2})^{2}$

11.

If $x^{2} + b x + c$ is factored to be $(x -)^{2}$ , what is the missing value of the factored form of this expression?

Compare your answer:

$\frac{b}{2}$

Notice that finding the value of $c$ helped to create a perfect square trinomial. This is called completing the square.

Building Character: Intellectual Humility

Two people stand at podiums, engaged in a debate. Both wear suits with rosette pins and speak animatedly. Speech bubbles with question marks and ellipses hover above them. Balloons and plants decorate the scene.

When you approach life with intellectual humility, you open your mind to learning. You are able to learn from opposing views and have more constructive discussions, even when you disagree.

Think about yourself. Are these statements true for you?

I question my own opinions, positions, and viewpoints because they could be wrong.
In the face of conflicting evidence, I am open to changing my opinions.

Don’t worry if none of these statements are true for you. Developing this trait takes time. Your first step starts today!

Self Check

Which value will create a perfect square trinomial when added to

x^2-14x + \_\_\_\_\_\_

?

196
49
-49
-7

Additional Resources

Completing the Square

When a number is added to a quadratic expression to make a perfect square trinomial, that is called completing the square.

Here is how to complete the square of a quadratic expression, $a x^{2} + b x + c$ .

How to Complete a Square of $x^{2} + b x$

Step 1 - Identify $b$ , the coefficient of $x$ .

Step 2 - Find half of $b$ and then square it. In other words, $(\frac{b}{2})^{2}$ .

Step 3 - Add the $(\frac{b}{2})^{2}$ to $x^{2} + b x$ .

Step 4 - Factor the perfect square trinomial, writing it as a binomial squared.

Example

Complete the square to make a perfect square trinomial. Then write the result as a binomial squared.

$x^{2} - 26 x$

Step 1 - Identify $b$ , the coefficient of $x$ .

The coefficient of $x$ is -26.

Step 2 - Find half of $b$ and then square it. In other words, $(\frac{b}{2})^{2}$ .

$(- \frac{26}{2})^{2} = (- 13)^{2} = 169$

Step 3 - Add the $(\frac{b}{2})^{2}$ to $x^{2} + b x$ .

$x^{2} - 26 x + 169$

Step 4 - Factor the perfect square trinomial, writing it as a binomial squared.

$(x - 13)^{2}$

Try it

Try It: Completing the Square

Complete the square to make a perfect square trinomial. Then, factor the trinomial.

$x^{2} + 8 x +____$

Here is how to complete the square.

Step 1 - Identify the coefficient of $x$ .

$b = 8$

Step 2 - Find $(\frac{b}{2})^{2}$ .

$(\frac{8}{2})^{2} = 4^{2} = 16$

Step 3 - Add the $(\frac{b}{2})^{2}$ to the $x^{2} + b x$ .

$x^{2} + 8 x + 16$

Step 4 - Factor the perfect square trinomial.

$(x + 4)^{2}$

9.2.2 Standard and Factored Forms of Perfect Squares

Activity

Additional Resources

Completing the Square

Try It: Completing the Square