SOLVING EQUATIONS WITH RADICALS ON BOTH SIDES WORKSHEET

Solve the following equations and check the answers :

Question 1 :

x = 52

Question 2 :

√-y = 3√7

Question 3 :

√(3x + 12) = 3√3

Question 4 :

√(x - 5) = 2√6

Question 5 :

√(3x - 4) √6

Question 6 :

√(x2 - 2) √(x + 4)

Question 7 :

√(x - 6) √(x + 9) - 3

Question 8 :

3√(x - 6) 3(2x - 9)

1. Answer :

x = 52

Square both sides.

(√x)= (52)2

x = 5222

x = 25(2)

x = 50

Check Answer :

√50 = 52 ?

√(5 ⋅ 5 ⋅ 2) = 5√2 ?

5√2 = 5√2 

Therefore, the solution is

x = 50

2. Answer :

√-y = 3√7

Square both sides.

(√-y)= (3√7)2 

-y = 32√72

-y = 9(7)

-y = 63

Multiply both sides by -1.

y = -63

Check Answer :

√[-(-63)] = 3√7 ?

√63 = 3√7 ?

√(3 ⋅ 3 ⋅ 7) = 3√7 ?

3√7 = 3√7 

Therefore, the solution is

y = -63

3. Answer :

√(3x + 12) = 3√3

Square both sides.

[√3x + 12)]2 = (3√3)2

3x + 12 = 32√32

3x + 12 = 9(3)

3x + 12 = 27

Subtract 12 from both sides.

3x = 15

Divide both sides by 3.

x = 5

Check Answer :

√(3(5) + 12) = 3√3 ?

√(15 + 12) = 3√3 ?

√27 = 3√3 ?

√(3 ⋅ 3 ⋅ 3) = 3√3 ?

3√2 = 3√3 

Therefore, the solution is

x = 5

4. Answer :

√(x - 5) = 2√6

Square both sides.

[√(x - 5)]2 = (2√6)2

x - 5 = 22√62

x - 5 = 4(6)

x - 5 = 24

Add 5 to both sides.

x = 29

Check Answer :

√(29 - 5) = 2√6 ?

√24 = 2√6 ?

√(2 ⋅ 2 ⋅ 2 ⋅ 3) = 2√6 ?

2√(⋅ 3) = 2√6 ?

2√6 = 2√6 

Therefore, the solution is

x = 29

5. Answer :

√(3x - 4) = √6

Square both sides.

[(√(3x - 4)]2 = (√6)2

3x - 4 = √62

3x - 4 = 6

Add 4 to both sides.

3x = 6 + 4

3x = 10

Divide both sides by 3.

x = 10/3

Check Answer :

√(3(10/3) - 4) √6 ?

√(10 - 4) √6 ?

√6 √6 

Therefore, the solution is

x = 10/3

6. Answer :

√(x2 - 2) √(x + 4)

Square both sides.

[√(x2 - 2)]= [√(x + 4)]2

x2 - 2 = x + 4

Subtract x and 4 from both sides.

x2 - x - 6 = 0

Solve by factoring.

(x + 2)(x - 3) = 0 

x + 2 = 0

x = -2

x - 3 = 0

x = 3

Check Answers :

x = -2 :

√((-2)2 - 2) √(-2 + 4) ?

√(4 - 2) √2 ?

√2 √2 

x = 3 :

√(32 - 2) √(3 + 4) ?

√(9 - 2) √7 ?

√7 √7 

Therefore, the solutions are

x = -2 and x = 3

7. Answer :

√(x - 6) √(x + 9) - 3

Square both sides.

[√(x - 6)]= [√(x + 9) - 3]2

x - 6 = [√(x + 9) - 3][√(x + 9) - 3]

x - 6 = [√(x + 9)]2 - 3√(x + 9) - 3√(x + 9) + 32

x - 6 = x + 9 - 6√(x + 9) + 9

x - 6 = x - 6√(x + 9) + 18

Subtract x from both sides.

-6 = -6√(x + 9) + 18

Subtract 18 from both sides.

-24 = -6√(x + 9)

Square both sides.

(-24)2 = [-6√(x + 9)]2

576 = (-6)2[√(x + 9)]2

576 = 36(x + 9)

Divide both sides by 576.

16 = x + 9

Subtract 9 from both sides.

7 = x

Check Answer :

√(7 - 6) = √(7 + 9) - 3 ?

√1 = √16 - 3 ?

1 = 4 - 3 ?

1 = 1 

Therefore, the solution is

x = 7

8. Answer :

3√(x - 6) 3(2x - 9)

Raise both sides to the 3rd power.

[3√(x - 6)][3(2x - 9)]3

x - 6 2x - 9

Subtract x from both sides.

-6 = x - 9

Add 9 to both sides.

3 = x

Check Answer :

3√(3 - 6) 3(2(3) - 9) ?

3√(-3) 3(6 - 9) ?

3√(-3) = 3(-3) 

Therefore, the solution is

x = 3

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