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5‐7 Completing the SquareDay 1

Objectives: *Solve equations by completing the square. *Rewrite functions by completing the square.

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Check Skills You'll Need:Simplify each expression

1. (x ‐ 3)(x ‐ 3) 2. (2x ‐ 1)(2x ‐ 1) 3. (x + 4)(x + 4)

4. ±√25 5. ±√48 6. ±√‐4 7. ±√9/16

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You can solve and equation in which one side is a perfect square trinomial by finding square roots.

Example #1: Solve x2 + 10x + 25 = 36

x2 + 10x + 25 = 36

(x + 5)2 = 36

x + 5 = ±6

x + 5 = 6 or x + 5 = ‐6

x = 1 x = ‐11

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If one side of an equation is not a perfect square trinomial, you can rewrite the constant term to get a perfect square trinomial. The process of finding the last term of a perfect square trinomial is called completing the square.

Use the following relationship to find the term that will complete the square.

x2 + bx + ( )2 = (x + )2b2

b2

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Example #2: Find the missing value to complete the square.

x2 ‐ 8x +

( )2 = ( )2 = 16

x2 ‐ 8x + 16

b2

‐82

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Example #3: Find the missing value to complete the square.

x2 + 7x +

( )2 = ( )2 = 12.25

x2 + 7x + 12.25

b2

72

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You can solve any quadratic equation by completing the square.

Example #4: Solve x2 ‐ 12x + 5 = 0

( )2 = 36 Find ( )2

x2 ‐ 12x = ‐5 Rewrite so all terms containing x are on one side.

x2 ‐ 12x + 36 = ‐5 + 36 Complete the square by adding 36 to each side.

(x ‐ 6)2 = 31 Factor the perfect square trinomial.

x ‐ 6 = ±√31 Find square roots

x = 6 ±√31 Solve for x.

b2

‐122

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Example #5: Solve each equation by completing the square.

a. x2 + 4x ‐ 4 = 0 b. x2 ‐ 2x ‐ 1 = 0

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By completing the square, you can solve equations that cannot bysolved by factoring, finding square roots, or graphing.

Example #6: Solve x2 ‐ 8x + 36 = 0

( )2 = 16

x2 ‐ 8x = ‐36

x2 ‐ 8x + 16 = ‐36 + 16

(x ‐ 4)2 = ‐20

x ‐ 4 = ±√‐20

x = 4 ± 2i√5

‐82

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Example #7: Write y = x2 + 6x + 2 in vertex form.

x2 + 6x + 2 = 0

( )2 = 9

x2 + 6x = ‐2

x2 + 6x + 9 = ‐2 + 9

(x + 3)2 = 7

(x + 3)2 ‐ 7 = 0

y = (x + 3)2 ‐ 7

What is the vertex of the function? (‐3, ‐7)

62

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Example #8: Write each equation in vertex form.

a. y = x2 ‐ 10x ‐ 2 b. y = x2 + 5x + 3

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Homework:page 285 (1 ‐ 18, 28, 31)