1. Section 4.6 Completing the Square

2. Perfect Trinomials:
When a perfect trinomial is factored, both binomials will be equal
The third term in a perfect trinomial is equal to the second term divided by 2 and then squared
The term in the binomial is equal to the second term divided by 2
When we CTS, we change the trinomial into a perfect trinomial
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3. What is “Completing the Square”
“Completing the Square” is a process that converts a quadratic function that can not be factored to a form that can be solved algebraically
Suppose we have the following trinomial
If we complete the square, we end up with the expression on the right
Can’t factor this…
We end up with two answers in radical form

4. How to Complete the Square
Bracket the first two terms!
Divide the second term by 2 and square it! Purpose: Make the expression in the bracket into a perfect square!
Take the negative square outside of the brackets!
The trinomial becomes two equal binomials
Now you can solve this equation by square rooting both sides:

5. Ex: Complete the Square and Solve for “x”
Bracket the first two terms!
Factor out any coefficient for x2
Divide the second term by 2 and square it!
Take the negative square outside of the brackets and multiply with coefficient in front!
The trinomial becomes two equal binomials
Solve for “x” by square rooting both sides

6. Ex: Solve for “x” by Completing the Square:

7. Ex: The sum of two numbers is 80. Their product is 1500
Ex: The sum of two numbers is 80. Their product is Find the numbers
There are two numbers, let ‘x’ and ‘y’ be the numbers
Gather Information
The sum is 80
The product is 1600
Substitute the first equation into the second one.
The two numbers are 30 and 80

8. Ex: The difference of two numbers is 10. The sum of their squares is 60. Find the numbers
There are two numbers, let ‘x’ and ‘y’ be the numbers
Gather Information
The difference is 10
The sum of their squares is 60
Substitute the first equation into the second one.
The two sets of numbers are: