SOLVING RADICAL EQUATIONS WORKSHEET

Solve the following rational equations.

1. √(2x - 1) = 5

2. √(x + 1) + 7 = 10

3. √(x + 2) = x - 4

4. √(3x - 5) = x - 5

5. x = √(6 - x)

6. x + √(6 - x) = 4

7. √(x² + 9x + 14) = x + 4

8. √(x - 4) - √x = 2

9. ³√(3x) - 9 = 0

10. ³√(x + 2) + 7 = 0

Answers

1. Answer :

√(2x - 1) = 5

Raise both sides to the power 2.

[√(2x - 1)]² = 5²

2x - 1 = 25

Add 1 to both sides.

2x = 26

Divide both sides by 2.

x = 13

Substitute x = 13.

√(2 ⋅ 13 - 1) = 5

√(26 - 1) = 5

√25 = 5

5 = 5 (true)

So, 5 is the solution for the given radical equation.

2. Answer :

√(x + 1) + 7 = 10

Subtract 7 from both sides.

√(x + 1) = 3

Raise both sides to the power 2.

[√(x + 1)]² = 3²

x + 1 = 9

Subtract 1 from both sides.

x = 8

Substitute x = 8.

√(8 + 1) + 7 = 10

√9 + 7 = 10

3 + 7 = 10

10 = 10 (true)

So, 8 is the solution for the given radical equation.

3. Answer :

√(x + 2) = x - 4

Raise both sides to the power 2.

[√(x + 2)]² = (x - 4)²

x + 2 = (x - 4)(x - 4)

x + 2 = x² - 4x - 4x + 16

x + 2 = x² - 8x + 16

Subtract x and 2 from both sides.

0 = x² - 9x + 14 

Factor and solve :

(x - 2)(x - 7) = 0

x -2 = 0 or x - 7 = 0

Substitute x = 2 and x = 7 in the given equation.

x = 2 does not satisfy the given equation

So, x = 7 is the only solution for the given radical equation.

4. Answer :

√(3x - 5) = x - 5

Raise both sides to the power 2.

[√(3x - 5)]² = (x - 5)²

3x - 5 =  (x - 5)(x - 5)

3x - 5 = x² - 5x - 5x + 25

3x - 5 = x² - 10x + 25

Subtract 3x from both sides and 5 to both sides.

0 = x² - 13x + 30

Factor and solve :

(x - 10)(x - 3) = 0

x - 10 = 0 or x - 3 = 0

Substitute x = 10 and x = 3 in the given equation.

x = 3 does not satisfy the original equation.

So, x = 10 is the only solution for the given radical equation.

5. Answer :

x = √(6 - x)

Raise both sides to the power 2.

x² = [√(6 - x)]²

x² = 6 - x

Add x to both sides and subtract 6 from both sides.

x² + x - 6 = 0

Factor and solve :

(x - 2)(x + 3) = 0

x - 2 = 0 or x + 3 = 0

Substitute x = 2 and x = -3 in the given equation.

x = -3 does not satisfy the original equation.

So, x = 2 is the only solution for the given radical equation.

6. Answer :

x + √(6 - x) = 4

Subtract x from both sides.

√(6 - x) = 4 - x

Raise both sides to the power 2.

[√(6 - x)]² = (4 - x)²

6 - x = (4 - x)(4 - x)

6 - x = 16 - 4x - 4x + x²

6 - x = 16 - 8x + x²

Add x to both sides and subtract 6 from both sides.

0 = 10 - 7x + x²

x² - 7x + 10 = 0

Factor and solve :

(x - 2)(x - 5) = 0

x - 2 = 0 or x - 5 = 0

Substitute x = 2 and x = 5 in the given equation.

x = 5 does not satisfy the original equation.

So, x = 2 is the only solution for the given radical equation.

7. Answer :

√(x² + 9x + 14) = x + 4

Raise both sides to the power 2.

[√(x² + 9x + 14)]² = (x + 4)²

x² + 9x + 14 = (x + 4)(x + 4)

x² + 9x + 14  =  x² + 4x + 4x + 16

x² + 9x + 14  =  x² + 8x + 16

Subtract x² from both sides. 

9x + 14 = 8x + 16

Subtract 8x from both sides. 

x + 14 = 16

Subtract 14 from both sides. 

x = 2

Substitute x = 2.

√(2² + 9(2) + 14) = 2 + 4

√(4 + 18 + 14) = 6

√36 = 6

6 = 6 (true)

So, 2 is the solution for the given radical equation.

8. Answer :

√(x - 4) - √x = 2

Raise both sides to the power 2.

[√(x - 4) - √(x)]² = 2²

Using the identity (a - b)² = a² - 2ab + b² on the left side,

[√(x - 4)]² - 2√(x - 4)√x + [√x]² = 4

x - 4 - 2√[x(x - 4)] + x = 4

2x - 4 - 2√(x²- 4x) = 4

-2√(x²- 4x) = 8 - 2x

Divide both sides by -2.

√(x²- 4x) = 4 - x

Raise both sides to the power 2.

[√(x²- 4x)]² = (4 - x)²

x²- 4x = (4 - x)(4 - x)

x²- 4x = 16 - 4x - 4x + x²

x²- 4x = 16 - 8x + x²

-4x = 16 - 8x

Add 8x to both sides.

4x = 16

Divide both sides by 4.

x = 4

Substitute x = 4.

√(4 - 4) - √4 = 2

0 - 2 = 2

-2 = 2 (false)

So, the given equation has no solution.

9. Answer :

³√(3x) - 9 = 0

Add 9 to both sides.

³√(3x) = 9

Raise both sides to the power 3.

[³√(3x)]³ = 9³

3x = 729

Divide both sides by 3.

x = 243

Substitute x = 243.

³√(3 ⋅ 243) - 9 = 0

³√729 - 9 = 0

9 - 9 = 0

0 = 0 (true)

So, 243 is the solution for the given radical equation.

10. Answer :

³√(x + 2) + 7 = 0

Subtract 7 from both sides.

³√(x + 2) = -7

Raise both sides to the power 3.

[³√(x + 2)]³ = (-7)³

x + 2 = -343

Subtract 2 from both sides.

x = -345

Substitute x = 4.

³√(-345 + 2) + 7 = 0

³√(-343) + 7 = 0

-7 + 7 = 0

0 = 0 (true)

So, -345 is the solution for the given radical equation.

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