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10.4: Solving Quadratic Equations Using the Method of Extraction of Roots

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The Method Of Extraction Of Roots

Extraction of Roots

Quadratic equations of the form x2K=0 can be solved by the method of extraction of roots by rewriting it in the form x2=K.

To solve x2=K, we are required to find some number, x, that when squared produces K. This number, x, must be a square root of K. If K is greater than zero, we know that it prossesses two square roots, K and K. We also know that

(K)2)=(K)(K)=K and (K)=(K)(K)=K

We now have two replacements for x that produce true statements when substitued into the equation. Thus, x=K and x=K are both solutions to x2=K. We use the notation x=±K to denote both the principal and secondary square roots.

The Nature of Solutions

Solutions of x2=K

For quadratic equations of the form x2=K,

  1. If K is greater than or equal to zero, the solutions are ±K.
  2. If K is negative, no real number solutions exist.
  3. If K is zero, the only solution is 0.

Sample Set A

Solve each of the following quadratic equations using the method of extraction of roots.

Example 10.4.1

x249=0 Rewrite x2=49x=±49x=±7

Check:

(7)2=49 Is this correct? (7)2=49 Is this correct? 49=49 Yes, this is correct. 49=49 Yes, this is correct. 

Example 10.4.2

25a2=36a2=3625a=±3625a=±65

Check:

25(65)2=36 Is this correct? 25(65)2=36 Is this correct? 25(3625)2=36 Is this correct? 25(3625)=36 Is this correct? 36=36Yes, this is correct. 36=36 Yes, this is correct. 

Example 10.4.3

4m232=04m2=32m2=324m2=8m=±8m=±22

Check:

4(22)2=32 Is this correct? 4(22)2=32 Is this correct? 4[22(2)2]=32 Is this correct? 4[(2)2(2)2]=32 Is this correct? 4[42]=32 Is this correct? 4[42]=32 Is this correct? 48=32 Is this correct? 48=32 Is this correct? 32=32 Yes, this is correct. 32=32 Yes, this is correct. 

Example 10.4.4

Solve 5x215y2z7=0 for x.

5x2=15y2z7 Divide both sides by 5x2=3y2z7x=±3y2z7x=±yz33z

Example 10.4.5

Calculator Problem:

Solve 14a2235=0. Round to the nearest hundredth.

14a2235=0 Rewrite 14a2=235 Divide both sides by 14a2=23514

Rounding to the nearest hundredth produces 4.10. We must be sure to insert the ± symbol.

a±4.10.

Example 10.4.6

k2=64k=±64

The radicand is negative so no real number solutions exist

Practice Set A

Solve each of the following quadratic equations using the method of extraction of roots.

Practice Problem 10.4.1

x2144=0

Answer

x=±12

Practice Problem 10.4.2

9y2121=0

Answer

y=±113

Practice Problem 10.4.3

6a2=108

Answer

a=±32

Practice Problem 10.4.4

Solve 4n2=24m2p8 for n.

Answer

n=±mp46

Practice Problem 10.4.5

Solve 5p2q2=45p2 for q

Answer

q=±3

Practice Problem 10.4.6

Solve 16m22206=0. Round to the nearest hundredth.

Answer

m=±11.74

Practice Problem 10.4.7

h2=100

Sample Set B

Solve each of the following quadratic equations using the method of extraction of roots.

Example 10.4.7

(x+2)2=81x+2=±81x+2=±9 Subtract 2 from both sides.x=2±9x=2+9andx=29x=7x=11

Example 10.4.8

(a+3)2=5a+3=±5 Subtract 3 from both sides a=3±5

Practice Set B

Solve each of the following quadratic equations using the method of extraction of roots.

Practice Problem 10.4.8

(a+6)2=64

Answer

a=2,14

Practice Problem 10.4.9

(m4)2=15

Answer

m=4±15

Practice Problem 10.4.10

(y7)2=49

Answer

y=0,14

Practice Problem 10.4.11

(k1)2=12

Answer

k=1±23

Practice Problem 10.4.12

(x11)2=0

Answer

x=11

Exercises

For the following problems, solve each of the quadratic equations using the method of extraction of roots.

Exercise 10.4.1

x2=36

Answer

x=±6

Exercise 10.4.2

x2=49

Exercise 10.4.3

a2=9

Answer

a=±3

Exercise 10.4.4

a2=4

Exercise 10.4.5

b2=1

Answer

b=±1

Exercise 10.4.6

a2=1

Exercise 10.4.7

x2=25

Answer

x=±5

Exercise 10.4.8

x2=81

Exercise 10.4.9

a2=5

Answer

a=±5

Exercise 10.4.10

a2=10

Exercise 10.4.11

b2=12

Answer

b=±23

Exercise 10.4.12

b2=6

Exercise 10.4.13

y2=3

Answer

y=±3

Exercise 10.4.14

y2=7

Exercise 10.4.15

a28=0

Answer

a=±22

Exercise 10.4.16

a23=0

Exercise 10.4.17

a25=0

Answer

a=±5

Exercise 10.4.18

y21=0

Exercise 10.4.19

x210=0

Answer

x=±10

Exercise 10.4.20

x211=0

Exercise 10.4.21

3x227=0

Answer

x=±3

Exercise 10.4.22

5b25=0

Exercise 10.4.23

2x2=50

Answer

x=±5

Exercise 10.4.24

4a2=40

Exercise 10.4.25

2x2=24

Answer

x=±23

For the following problems, solve for the indicated variable.

Exercise 10.4.26

x2=4a2, for x

Exercise 10.4.27

x2=9b2, for x

Answer

x=±3b

Exercise 10.4.28

a2=25c2, for a

Exercise 10.4.29

k2=m2n2, for k.

Answer

k=±mn

Exercise 10.4.30

k2=p2q2r2, for k

Exercise 10.4.31

2y2=2a2n2, for y

Answer

y=±an

Exercise 10.4.32

9y2=27x2z4, for y.

Exercise 10.4.33

x2z2=0, for x

Answer

x=±z

Exercise 10.4.34

x2z2=0, for z

Exercise 10.4.35

5a210b2=0, for a

Answer

a=b2,b2

For the following problems, solve each of the quadratic equations using the method of extraction of roots.

Exercise 10.4.36

(x1)2=4

Exercise 10.4.37

(x2)2=9

Answer

x=5,1

Exercise 10.4.38

(x3)2=25

Exercise 10.4.39

(a5)2=36

Answer

x=11,1

Exercise 10.4.40

(a+3)2=49

Exercise 10.4.41

(a+9)2=1

Answer

a=8,10

Exercise 10.4.42

(a6)2=3

Exercise 10.4.43

(x+4)2=5

Answer

a=4±5

Exercise 10.4.44

(b+6)2=7

Exercise 10.4.45

(x+1)2=a, for x.

Answer

x=1±a

Exercise 10.4.46

(y+5)2=b, for y

Exercise 10.4.47

(y+2)2=a2, for y

Answer

y=2±a

Exercise 10.4.48

(x+10)2=c2, for x

Exercise 10.4.49

(xa)2=b2, for x

Answer

x=a±b

Exercise 10.4.50

(x+c)2=a2, for x

Calculator Problems

For the following problems, round each result to the nearest hundredth.

Exercise 10.4.51

8a2168=0

Answer

a=±4.58

Exercise 10.4.52

6m25=0

Exercise 10.4.53

0.03y2=1.6

Answer

y=±7.30

Exercise 10.4.54

0.048x2=2.01

Exercise 10.4.55

1.001x20.999=0

Answer

x=±1.00

Exercises For Review

Exercise 10.4.56

Graph the linear inequality 3(x+2)<2(3x+4)

A horizontal line with arrows on both ends.

Exercise 10.4.57

Solve the fractional equation: x1x+4=x+3x1

Answer

x=119

Exercise 10.4.58

Find the product: 32x3y52x3y3

Exercise 10.4.59

Solve x24x=0

Answer

x=0,4

Exercise 10.4.60

Solve y28y=12


This page titled 10.4: Solving Quadratic Equations Using the Method of Extraction of Roots is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Denny Burzynski & Wade Ellis, Jr. (OpenStax CNX) .

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