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10.2 Solve Quadratic Equations by Completing the Square

Adding the term that turns a squared variable plus its middle term into a perfect square trinomial, dividing out any leading coefficient first.

Elementary Algebra 2e, 10.2 Solve Quadratic Equations by Completing the Square

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52 exercises

10.2 Exercise 1

Before you get started, take this readiness quiz. If you miss a problem, go back to the section listed and review the material.

Simplify (x+12)2\left(x + 12\right)^{2}.
If you missed this problem, review the reference.

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10.2 Exercise 2

Factor y2−18y+81y^{2} - 18 y + 81.
If you missed this problem, review the reference.

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10.2 Exercise 3

Factor 5n2+40n+805 n^{2} + 40 n + 80.
If you missed this problem, review the reference.

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10.2 Exercise 5

Complete the square to make a perfect square trinomial. Write the result as a binomial square.

y2+12yy^{2} + 12 y

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10.2 Exercise 6

Complete the square to make a perfect square trinomial. Write the result as a binomial square.

z2+8zz^{2} + 8 z

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10.2 Exercise 8

Complete the square to make a perfect square trinomial. Write the result as a binomial square.

a2−20aa^{2} - 20 a

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10.2 Exercise 9

Complete the square to make a perfect square trinomial. Write the result as a binomial square.

b2−4bb^{2} - 4 b

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10.2 Exercise 11

Complete the square to make a perfect square trinomial. Write the result as a binomial square.

m2−5mm^{2} - 5 m

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10.2 Exercise 12

Complete the square to make a perfect square trinomial. Write the result as a binomial square.

n2+13nn^{2} + 13 n

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10.2 Exercise 14

Complete the square to make a perfect square trinomial. Write the result as a binomial square.

p2+14pp^{2} + \frac{1}{4} p

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10.2 Exercise 15

Complete the square to make a perfect square trinomial. Write the result as a binomial square.

q2−23qq^{2} - \frac{2}{3} q

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10.2 Exercise 17

Solve c2+4c=5c^{2} + 4 c = 5 by completing the square.

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10.2 Exercise 18

Solve d2+10d=−9d^{2} + 10 d = −9 by completing the square.

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10.2 Exercise 20

Solve r2−4r=12r^{2} - 4 r = 12 by completing the square.

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10.2 Exercise 21

Solve t2−10t=11t^{2} - 10 t = 11 by completing the square.

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10.2 Exercise 23

Solve y2−10y=−35y^{2} - 10 y = −35 by completing the square.

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10.2 Exercise 24

Solve z2+8z=−19z^{2} + 8 z = −19 by completing the square.

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10.2 Exercise 26

Solve x2−16x=−16x^{2} - 16 x = −16 by completing the square.

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10.2 Exercise 27

Solve y2+8y=11y^{2} + 8 y = 11 by completing the square.

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10.2 Exercise 29

Solve a2+4a+9=30a^{2} + 4 a + 9 = 30 by completing the square.

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10.2 Exercise 30

Solve b2+8b−4=16b^{2} + 8 b - 4 = 16 by completing the square.

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10.2 Exercise 32

Solve p2=5p+9p^{2} = 5 p + 9 by completing the square.

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10.2 Exercise 33

Solve q2=7q−3q^{2} = 7 q - 3 by completing the square.

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10.2 Exercise 35

Solve (c−2)(c+8)=7\left(c - 2\right) \left(c + 8\right) = 7 by completing the square.

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10.2 Exercise 36

Solve (d−7)(d+3)=56\left(d - 7\right) \left(d + 3\right) = 56 by completing the square.

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10.2 Exercise 38

Solve 2m2+16m−8=02 m^{2} + 16 m - 8 = 0 by completing the square.

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10.2 Exercise 39

Solve 4n2−24n−56=84 n^{2} - 24 n - 56 = 8 by completing the square.

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10.2 Exercise 41

Solve 3r2−2r=213 r^{2} - 2 r = 21 by completing the square.

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10.2 Exercise 42

Solve 4t2+2t=204 t^{2} + 2 t = 20 by completing the square.

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10.2 Exercise 44

Solve 4x2+3x=124 x^{2} + 3 x = 12 by completing the square.

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10.2 Exercise 45

Solve 5y2+3y=105 y^{2} + 3 y = 10 by completing the square.

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10.2 Exercise 46

Complete the square to make a perfect square trinomial. Then, write the result as a binomial squared.

a2+10aa^{2} + 10 a

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10.2 Exercise 48

Complete the square to make a perfect square trinomial. Then, write the result as a binomial squared.

m2+18mm^{2} + 18 m

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10.2 Exercise 50

Complete the square to make a perfect square trinomial. Then, write the result as a binomial squared.

m2−24mm^{2} - 24 m

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10.2 Exercise 52

Complete the square to make a perfect square trinomial. Then, write the result as a binomial squared.

p2−22pp^{2} - 22 p

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10.2 Exercise 54

Complete the square to make a perfect square trinomial. Then, write the result as a binomial squared.

x2−9xx^{2} - 9 x

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10.2 Exercise 56

Complete the square to make a perfect square trinomial. Then, write the result as a binomial squared.

p2−13pp^{2} - \frac{1}{3} p

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10.2 Exercise 58

Solve by completing the square.

v2+6v=40v^{2} + 6 v = 40

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10.2 Exercise 60

Solve by completing the square.

u2+2u=3u^{2} + 2 u = 3

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10.2 Exercise 62

Solve by completing the square.

c2−12c=13c^{2} - 12 c = 13

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10.2 Exercise 64

Solve by completing the square.

x2−20x=21x^{2} - 20 x = 21

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10.2 Exercise 66

Solve by completing the square.

m2+4m=−44m^{2} + 4 m = −44

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10.2 Exercise 68

Solve by completing the square.

r2+6r=−11r^{2} + 6 r = −11

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10.2 Exercise 70

Solve by completing the square.

a2−10a=−5a^{2} - 10 a = −5

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10.2 Exercise 72

Solve by completing the square.

u2−14u+12=−1u^{2} - 14 u + 12 = −1

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10.2 Exercise 74

Solve by completing the square.

v2=9v+2v^{2} = 9 v + 2

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10.2 Exercise 76

Solve by completing the square.

(x+6)(x−2)=9\left(x + 6\right) \left(x - 2\right) = 9

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10.2 Exercise 78

Solve by completing the square.

3m2+30m−27=63 m^{2} + 30 m - 27 = 6

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10.2 Exercise 80

Solve by completing the square.

2c2+c=62 c^{2} + c = 6

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10.2 Exercise 82

Solve by completing the square.

2p2+7p=142 p^{2} + 7 p = 14

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10.2 Exercise 84

Rafi is designing a rectangular playground to have an area of 320 square feet. He wants one side of the playground to be four feet longer than the other side. Solve the equation p2+4p=320p^{2} + 4 p = 320 for pp, the length of one side of the playground. What is the length of the other side?

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10.2 Exercise 86

Solve the equation x2+10x=−25x^{2} + 10 x = −25 ⓐ by using the Square Root Property and ⓑ by completing the square. ⓒ Which method do you prefer? Why?

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Adapted from Elementary Algebra 2e © 2020 Rice University. Licensed under CC BY 4.0. Source: Elementary Algebra 2e. Imported at revision 77a933c.