• completing the square
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.
Equation with Rational Roots
x = –7 + 8 or x = –7 – 8
x=1
x = –15
Write as two equations.
Solve each equation.
Answer: The solution set is {–15, 1}.
Solve x 2 – 16x + 64 = 25 by using the Square Root
Property.
A. {–1, 9}
B. {11, 21}
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B
A
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A
B
C
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D
D
D. {–13, –3}
C
C. {3, 13}
A.
B.
C.
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D.
Equation with Irrational Roots
Solve x 2 – 4x + 4 = 13 by using the Square Root
Property.
Original equation
Factor the perfect square
trinomial.
Square Root Property
Add 2 to each side.
Write as two equations.
Use a calculator.
Equation with Irrational Roots
Answer: The exact solutions of this equation are
The approximate
solutions are 5.61 and –1.61.
Solve x 2 – 4x + 4 = 8 by using the Square Root
Property.
A.
B.
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B
A
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A
B
C
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D
D
D.
A.
B.
C.
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D.
C
C.
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.
Step 2
Square the result of Step 1.
62 = 36
Step 3
Add the result of Step 2 to
x 2 + 12x.
x 2 + 12x + 36
Answer: The trinomial x2 + 12x + 36 can be written
as (x + 6)2.
Find the value of c that makes x2 + 6x + c a perfect
square. Then write the trinomial as a perfect
square.
A. 9; (x + 3)2
A
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B
D. 36; (x – 6)2
A
B
C
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D
D
C. 9; (x – 3)2
A.
B.
C.
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D.
C
B. 36; (x + 6)2
Solve an Equation by Completing the Square
Solve x2 + 4x – 12 = 0 by completing the square.
x2 + 4x – 12 = 0
x2 + 4x = 12
Notice that x2 + 4x – 12 is
not a perfect square.
Rewrite so the left side is
of the form x2 + bx.
x2 + 4x + 4 = 12 + 4
add 4 to
each side.
(x + 2)2 = 16
Write the left side as a
perfect square by factoring.
Solve an Equation by Completing the Square
x+2 =±4
x =– 2 ± 4
x = –2 + 4 or x = –2 – 4
x=2
x = –6
Square Root Property
Subtract 2 from each side.
Write as two equations.
Solve each equation.
Answer: The solution set is {–6, 2}.
Solve x2 + 6x + 8 = 0 by completing the square.
A.
B.
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B
A
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A
B
C
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D
D
D.
C
C.
A.
B.
C.
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D.
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
Equation with Imaginary Solutions
Subtract 2 from each side.
Solve x 2 + 4x + 5 = 0 by completing the square.
A.
B.
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B
A
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A
B
C
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D
D
D.
A.
B.
C.
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D.
C
C.
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.
Equation with a ≠ 1
Write the left side as a
perfect square by factoring.
Simplify the right side.
Square Root Property
Equation with a ≠ 1
or
x=1
Answer:
Write as two equations.
Solve each equation.
Solve 2x2 + 11x + 15 = 0 by completing the square.
A.
B.
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A
B
C
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D
D
A
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B
D.
C
C.
A.
B.
C.
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D.