Completing the square

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Completing the square
How would you solve a problem like this
Take the square root of both sides

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How would you solve a problem like this
Notice The trinomial is a perfect square.
Now its the same problem as the last slide
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What about a harder problem
This trinomial is not a perfect square.
You can make the trinomial a perfect square by
adding 2
Now you have an easier problem again
This is COMPLETING THE SQUARE
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COMPLETING THE SQUARE
You begin with a polynomial that is not a perfect
square.
Step 1 add (or subtract) a number to make the
polynomial into a perfect square.
Step 2 Factor the polynomial, and use square
roots to solve.
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How do you decide what to add?
What goes in the blank to make this a perfect
square?.
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How do you decide what to add?
If you start with
Notice theres no Coefficient on x2
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EXAMPLE 2
Solve a quadratic equation
Solve x2 16x 15 by completing the square.
SOLUTION
x2 16x 15
Write original equation.
x2 16x ( 8)2 15 ( 8)2
(x 8)2 15 ( 8)2
Write left side as the square of a binomial.
(x 8)2 49
Simplify the right side.
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EXAMPLE 2
x 8 7
Take square roots of each side.
x 8 7
Add 8 to each side.
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EXAMPLE 2
Standardized Test Practice
CHECK
You can check the solutions in the original
equation.
If x 1
If x 15
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EXAMPLE 3
Solve a quadratic equation in standard form
Solve 2x2 20x 8 0 by completing the square.
SOLUTION
2x2 20x 8 0
Write original equation.
2x2 20x 8
Add 8 to each side.
x2 10x 4
Divide each side by 2.
x2 10x 52 4 52
(x 5)2 29
Write left side as the square of a binomial.
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EXAMPLE 3
Solve a quadratic equation in standard form
Take square roots of each side.
Subtract 5 from each side.
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for Examples 2 and 3
GUIDED PRACTICE
4. x2 2x 3
SOLUTION
x2 2x 3
Write original equation.
x2 2x ( 1)2 3 ( 1)2
(x 1)2 3 ( 1)2
Write left side as the square of a binomial.
(x 1)2 4
Simplify the right side.
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for Examples 2 and 3
GUIDED PRACTICE
x 1 2
Take square roots of each side.
Add 8 to each side.
x 1 2
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for Examples 2 and 3
GUIDED PRACTICE
CHECK
You can check the solutions in the original
equation.
If x 3
If x 1
( 1)2 2( 1) 3
(3)2 2(3) 3
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for Examples 2 and 3
GUIDED PRACTICE
6. 3g2 24g 27 0
SOLUTION
3g2 24g 27 0
Write original equation.
3g2 24x ?27
Subtract 27 to each side.
g2 8g 9
Divide each side by 3.
g2 8g (4)2 9 (4)2
(g 4)2 6
Write left side as the square of a binomial.
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for Examples 2 and 3
GUIDED PRACTICE
Take square roots of each side.
Subtract 4 from each side.