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Equation with Rational Roots
Solve x
2 + 14x + 49 = 64 by using the Square Root Property.
Original equation
Factor the perfect square trinomial.
Square Root Property
Subtract 7 from each side.
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Equation with Rational Roots
x = 1 x = –15 Solve each equation.
Answer: The solution set is {–15, 1}.
x = –7 + 8 or x = –7 – 8 Write as two equations.
Check: Substitute both values into the original equation.
x
2 + 14x + 49 = 64 x
2 + 14x + 49 = 64??
1
2 + 14(1) + 49 = 64 (–15)
2 + 14(–15) + 49 = 64??
1 + 14 + 49 = 64 225 + (–210) + 49 = 64
64 = 64 64 = 64
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Equation with Irrational Roots
Solve x
2 – 4x + 4 = 13 by using the Square Root Property.
Square Root Property
Original equation
Factor the perfect square trinomial.
Add 2 to each side.
Write as two equations.
Use a calculator.
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Equation with Irrational Roots
x
2 – 4x + 4 = 13 Original equation
x
2 – 4x – 9 = 0 Subtract 13 from each side.
y = x
2 – 4x – 9 Related quadratic function
Answer: The exact solutions of this equation are The approximate solutions are 5.61 and –1.61. Check these results by finding and graphing the related quadratic function.
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Equation with Irrational Roots
Check Use the ZERO function of a graphing calculator. The approximate zeros of the related function are –1.61 and 5.61.
![Page 6: Example 1 Equation with Rational Roots Solve x 2 + 14x + 49 = 64 by using the Square Root Property. Original equation Factor the perfect square trinomial](https://reader031.vdocuments.site/reader031/viewer/2022013115/56649f045503460f94c1818f/html5/thumbnails/6.jpg)
A. A
B. B
C. C
D. D
Solve x
2 – 4x + 4 = 8 by using the Square Root Property.
A.
B.
C.
D.
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Complete the Square
Find the value of c that makes x
2 + 12x + c a perfect square. Then write the trinomial as a perfect square.
Step 1 Find one half of 12.
Answer: The trinomial x2 + 12x + 36 can be written as (x + 6)2.
Step 2 Square the result of Step 1. 62 = 36
Step 3 Add the result of Step 2 to x
2 + 12x + 36x
2 + 12x.
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A. A
B. B
C. C
D. D
A. 9; (x + 3)2
B. 36; (x + 6)2
C. 9; (x – 3)2
D. 36; (x – 6)2
Find the value of c that makes x2 + 6x + c a perfect square. Then write the trinomial as a perfect square.
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Solve an Equation by Completing the Square
Solve x2 + 4x – 12 = 0 by completing the square.
x2 + 4x – 12 = 0 Notice that x2 + 4x – 12 is not a perfect square.
x2 + 4x = 12Rewrite so
the left side is of the form x2 + bx.
x2 + 4x + 4 = 12 + 4
add 4 to
each side. (x + 2)2 = 16Write the
left side as a perfect square by factoring.
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Solve an Equation by Completing the Square
x + 2 = ± 4 Square Root Property
Answer: The solution set is {–6, 2}.
x = – 2 ± 4Subtract 2
from each side.
x = –2 + 4 or x = –2 – 4 Write as two equations.
x = 2 x = –6 Solve each equation.
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Equation with a ≠ 1
Solve 3x2 – 2x – 1 = 0 by completing the square.
3x2 – 2x – 1 = 0 Notice that 3x2 – 2x – 1 is not a perfect square.Divide by the coefficient of the quadratic term, 3.
Add to each side.
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Equation with a ≠ 1
Write the left side as a perfect square by factoring. Simplify the right side.
Square Root Property
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Equation with a ≠ 1
Answer:
x = 1 Solve each equation.
or Write as two equations.
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A. A
B. B
C. C
D. D
Solve 2x2 + 11x + 15 = 0 by completing the square.
A.
B.
C.
D.
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Equation with Imaginary Solutions
Solve x
2 + 4x + 11 = 0 by completing the square.
Notice that x
2 + 4x + 11 is not a perfect square.
Rewrite so the left side is of the form x
2 + bx.
Since , add 4 to each side.
Write the left side as a perfect square.
Square Root Property
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Equation with Imaginary Solutions
Subtract 2 from each side.