10.4: Solving Quadratic Equations Using the Method of Extraction of Roots
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The Method Of Extraction Of Roots
Quadratic equations of the form x2−K=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
For quadratic equations of the form x2=K,
- If K is greater than or equal to zero, the solutions are ±√K.
- If K is negative, no real number solutions exist.
- 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.
x2−49=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.
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.
4m2−32=04m2=32m2=324m2=8m=±√8m=±2√2
Check:
4(2√2)2=32 Is this correct? 4(−2√2)2=32 Is this correct? 4[22(√2)2]=32 Is this correct? 4[(−2)2(√2)2]=32 Is this correct? 4[4⋅2]=32 Is this correct? 4[4⋅2]=32 Is this correct? 4⋅8=32 Is this correct? 4⋅8=32 Is this correct? 32=32 Yes, this is correct. 32=32 Yes, this is correct.
Solve 5x2−15y2z7=0 for x.
5x2=15y2z7 Divide both sides by 5x2=3y2z7x=±√3y2z7x=±yz3√3z
Calculator Problem:
Solve 14a2−235=0. Round to the nearest hundredth.
14a2−235=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.
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.
x2−144=0
- Answer
-
x=±12
9y2−121=0
- Answer
-
y=±113
6a2=108
- Answer
-
a=±3√2
Solve 4n2=24m2p8 for n.
- Answer
-
n=±mp4√6
Solve 5p2q2=45p2 for q
- Answer
-
q=±3
Solve 16m2−2206=0. Round to the nearest hundredth.
- Answer
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m=±11.74
h2=−100
Sample Set B
Solve each of the following quadratic equations using the method of extraction of roots.
(x+2)2=81x+2=±√81x+2=±9 Subtract 2 from both sides.x=−2±9x=−2+9andx=−2−9x=7x=−11
(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.
(a+6)2=64
- Answer
-
a=2,−14
(m−4)2=15
- Answer
-
m=4±√15
(y−7)2=49
- Answer
-
y=0,14
(k−1)2=12
- Answer
-
k=1±2√3
(x−11)2=0
- Answer
-
x=11
Exercises
For the following problems, solve each of the quadratic equations using the method of extraction of roots.
x2=36
- Answer
-
x=±6
x2=49
a2=9
- Answer
-
a=±3
a2=4
b2=1
- Answer
-
b=±1
a2=1
x2=25
- Answer
-
x=±5
x2=81
a2=5
- Answer
-
a=±√5
a2=10
b2=12
- Answer
-
b=±2√3
b2=6
y2=3
- Answer
-
y=±√3
y2=7
a2−8=0
- Answer
-
a=±2√2
a2−3=0
a2−5=0
- Answer
-
a=±√5
y2−1=0
x2−10=0
- Answer
-
x=±√10
x2−11=0
3x2−27=0
- Answer
-
x=±3
5b2−5=0
2x2=50
- Answer
-
x=±5
4a2=40
2x2=24
- Answer
-
x=±2√3
For the following problems, solve for the indicated variable.
x2=4a2, for x
x2=9b2, for x
- Answer
-
x=±3b
a2=25c2, for a
k2=m2n2, for k.
- Answer
-
k=±mn
k2=p2q2r2, for k
2y2=2a2n2, for y
- Answer
-
y=±an
9y2=27x2z4, for y.
x2−z2=0, for x
- Answer
-
x=±z
x2−z2=0, for z
5a2−10b2=0, for a
- Answer
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a=b√2,−b√2
For the following problems, solve each of the quadratic equations using the method of extraction of roots.
(x−1)2=4
(x−2)2=9
- Answer
-
x=5,−1
(x−3)2=25
(a−5)2=36
- Answer
-
x=11,−1
(a+3)2=49
(a+9)2=1
- Answer
-
a=−8,−10
(a−6)2=3
(x+4)2=5
- Answer
-
a=−4±√5
(b+6)2=7
(x+1)2=a, for x.
- Answer
-
x=−1±√a
(y+5)2=b, for y
(y+2)2=a2, for y
- Answer
-
y=−2±a
(x+10)2=c2, for x
(x−a)2=b2, for x
- Answer
-
x=a±b
(x+c)2=a2, for x
Calculator Problems
For the following problems, round each result to the nearest hundredth.
8a2−168=0
- Answer
-
a=±4.58
6m2−5=0
0.03y2=1.6
- Answer
-
y=±7.30
0.048x2=2.01
1.001x2−0.999=0
- Answer
-
x=±1.00
Exercises For Review
Graph the linear inequality 3(x+2)<2(3x+4)
Solve the fractional equation: x−1x+4=x+3x−1
- Answer
-
x=−119
Find the product: √32x3y5√2x3y3
Solve x2−4x=0
- Answer
-
x=0,4
Solve y2−8y=−12