1. Simplify each expression.
Completing the Square
ALGEBRA 2 LESSON 5-7
(For help, go to Lessons 5-1 and page 262.) 9 16
1. (x – 3)(x – 3) 2. (2x – 1)(2x – 1)
3. (x + 4)(x + 4) – 3 4. ± 25
5. ± ± –4
7. ±
Simplify each expression.
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2. 1. (x – 3)(x – 3) = x2 – 2(3)x + 32 = x2 – 6x + 9
Completing the Square
ALGEBRA 2 LESSON 5-7
Solutions 9 16 3 4
1. (x – 3)(x – 3) = x2 – 2(3)x + 32 = x2 – 6x + 9
2. (2x – 1)(2x – 1) = (2x)2 – 2(1)(2x) + 12 = 4x2 – 4x + 1
3. (x + 4)(x + 4) – 3 = x2 + 2(4)x + 42 – 3 = x2 + 8x + 16 – 3 = x2 + 8x + 13
4. ± = ±5
5. ± 48 = ± • 3 = ±4 3
6. ± –4 = ± –1 • 4 = ±2i
7. ± = ± = ±
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3. (x – 6)2 = 9 Factor the trinomial.
Completing the Square
ALGEBRA 2 LESSON 5-7
Solve x2 – 12x + 36 = 9.
x2 – 12x + 36 = 9
(x – 6)2 = 9 Factor the trinomial.
x – 6 = ±3 Find the square root of each side.
x – 6 = 3 or x – 6 = –3 Solve for x.
x = 9 or
x = 3
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5. Find the missing value to complete the square: x2 + 20x + .
Completing the Square
ALGEBRA 2 LESSON 5-7
Find the missing value to complete the square: x2 + 20x
= = 100 Find Substitute 20 for b. b 2 20
x2 + 20x Complete the square.
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7. x2 + 4x = –1 Rewrite so all terms containing x are on one side.
Completing the Square
ALGEBRA 2 LESSON 5-7
Solve x2 + 4x + 1 = 0.
= 4 Find 4 2 b
x2 + 4x = –1 Rewrite so all terms containing x are on one side.
x2 + 4x + 4 = –1 + 4 Complete the square by adding 4 to each side.
(x + 2)2 = 3 Factor the perfect square trinomal.
x + 2 = ± 3 Find the square root of each side.
x = –2 ± 3 Solve for x.
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8. Completing the Square (continued) Check:
ALGEBRA 2 LESSON 5-7
(continued)
Check:
(–2)2 – 2(–2 3) + ( 3)2 – 8 –
(– )2 + 4(– ) + 1
x2 + 4x + 1
0 = 0
4 – –
(4 + 3 – 8 + 1) + (– )
(–2)2 + 2(–2 3) + ( 3)2 + (–8)
(–2 – 3)2 + 4(–2 – ) + 1
– 8 –
(4 + 3 – 8 + 1) + ( – )
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10. Rewrite so all terms containing x are on one side.
Completing the Square
ALGEBRA 2 LESSON 5-7
Solve x2 + 6x + 12 = 0. 6 2 b
= 9
Find
x2 + 6x = –12
Rewrite so all terms containing x are on one side.
x2 + 6x + 9 = –12 + 9
Complete the square by adding 9 to each side.
(x + 3)2 = –3
Factor the perfect square trinomial.
x + 3 = ± –3
Find the square root of each side.
x = –3 ± –3
Solve for x.
= –3 ± i 3
Simplify.
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12. = Completing the Square Solve 2x2 + 7x – 1 = 0. x2 + x – = 0
ALGEBRA 2 LESSON 5-7
Solve 2x2 + 7x – 1 = 0.
x x – = 0 7 2 1
Divide each side by 2.
x x = 1 2 7
Rewrite so all terms containing x are on one side. = 7 2 49 16
Find b
x x = + 7 2 49 16 1
Complete the square by adding to each side.
x = 7 4 57 16
Factor the perfect square trinomial. 2 7 4 57
x + = ±
Find the square root of each side.
x = – ± 7 4
Solve for x. 57
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14. Write y = x2 + 5x + 2 in vertex form.
Completing the Square
ALGEBRA 2 LESSON 5-7
Write y = x2 + 5x + 2 in vertex form.
y = x2 + 5x + 2
= x2 + 5x – 5 2
Complete the square. Add and subtract
on the right side.
= x – 25 4 5 2
Factor the perfect square trinomial.
= x – 5 2 17 4
Simplify.
The vertex form is y = x – . 5 2 17 4
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16. Homework
P. 281 # 13 – 33 eoo