SECTION 4.6COMPLETING THE SQUARE
PERFECT TRINOMIALS:
When a perfect trinomial is factored, both binomials will be equal
The third term in a perfect trinomial is equal to the secondterm divided by 2 and then squared
The term in the binomial is equal to the second term dividedby 2
When we CTS, we change the trinomial into a perfecttrinomial
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WHAT IS “COMPLETING THE SQUARE”
“Completing the Square” is a process that converts aquadratic function that can not be factored to a form thatcan be solved algebraically
Suppose we have the following trinomial
If we complete the square, we end up with the expressionon the right
Can’t factor this…
We end up with two answersin radical form
HOW TO COMPLETE THE SQUARE
Bracket the first two terms!
Divide the second term by 2 andsquare it!Purpose: Make the expression in thebracket into a perfect square!
Take the negative square outsideof the brackets!
The trinomial becomes twoequal binomials
Now you can solve this equation bysquare rooting both sides:
EX: COMPLETE THE SQUARE AND SOLVE FOR “X”
Bracket the first two terms!
Divide the second term by 2 andsquare it!
Take the negative squareoutside of the brackets andmultiply with coefficient infront!
The trinomial becomes twoequal binomials
Solve for “x” by squarerooting both sides
Factor out any coefficient for x2
Ex: The sum of two numbers is 80. Their product is1600. Find the numbers
GatherInformation
There are two numbers, let ‘x’ and ‘y’be the numbers
The sum is 80
The product is 1600
Substitute the first equationinto the second one.
The two numbersare 30 and 80
Ex: The difference of two numbers is 10. The sum oftheir squares is 60. Find the numbers
Gather Information
There are two numbers, let ‘x’ and ‘y’be the numbers
The difference is 10
The sum of theirsquares is 60
Substitute the first equationinto the second one.
The two sets of numbersare:
Find the vertex because:
The vertex is the minimum
1. Complete the square
The minimumoccurs when x =-5
HOMEWORK: