HOW TO SOLVE RADICAL EQUATIONS WITH RADICALS ON BOTH SIDES

Solve the following equations and check the answers :

Example 1 :

√x = 5√2

Solution :

√x = 5√2

Square both sides.

(√x)² = (5√2)²

x = 5²√2²

x = 25(2)

x = 50

Check Answer :

√50 = 5√2 ?

√(5 ⋅ 5 ⋅ 2) = 5√2 ?

5√2 = 5√2 ✓

Therefore, the solution is

x = 50

Example 2 :

√-y = 3√7 

Solution :

√-y = 3√7

Square both sides.

(√-y)² = (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

Example 3 :

√(3x + 12) = 3√3

Solution :

√(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

Example 4 :

√(x - 5) = 2√6

Solution :

√(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√(2 ⋅ 3) = 2√6 ?

2√6 = 2√6 ✓

Therefore, the solution is

x = 29

Example 5 :

√(3x - 4) = √6

Solution :

√(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

Example 6 :

√(x² - 2) = √(x + 4)

Solution :

√(x² - 2) = √(x + 4)

Square both sides.

[√(x² - 2)]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 

Check Answers :

Therefore, the solutions are

x = -2 and x = 3

Example 7 :

√(x - 6) = √(x + 9) - 3

Solution :

√(x - 6) = √(x + 9) - 3

Square both sides.

[√(x - 6)]² = [√(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

Example 8 :

³√(x - 6) = ³√(2x - 9)

Solution :

³√(x - 6) = ³√(2x - 9)

Raise both sides to the 3^rd power.

[³√(x - 6)]³ = [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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