Worksheet: Radical Equations

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In this worksheet, we will practice solving equations that include radical expressions, where the variable is the radicand, by raising each side to a power.

Q1:

The natural way to solve √ 𝑥 + 9 − 2 𝑥 = 3 introduces an extraneous solution. What is that extraneous solution?

  • A 𝑥 = − 3 4
  • B 𝑥 = − 2
  • C 𝑥 = 1 1 7
  • D 𝑥 = − 1 1 4
  • E 𝑥 = 3

Q2:

Solve √ 𝑥 − 5 = − 1 .

  • A 𝑥 = 6
  • B 𝑥 = 3 6
  • C 𝑥 = 4
  • D 𝑥 = 1 6

Q3:

Solve √ 𝑥 + 4 + 2 = 7 .

  • A 𝑥 = 2 9
  • B 𝑥 = 4 3
  • C 𝑥 = 1
  • D 𝑥 = 2 1
  • E 𝑥 = 9

Q4:

Find the value of 𝑥 given that 3 √ 𝑥 = − √ 3 6 .

Q5:

Find the solution set of the equation − √ 𝑥 + 1 = 7 3 , given that 𝑥 ∈ ℝ .

  • A { 5 1 2 }
  • B { − 5 1 2 }
  • C { − 6 }
  • D { − 2 1 6 }
  • E { 2 1 6 }

Q6:

Find the value of 𝑥 for which √ 1 2 2 5 = √ 4 0 0 + √ 𝑥 .

Q7:

Given that 3 √ 𝑥 + 3 4 = 4 , find the value of 𝑥 .

  • A 27 000
  • B324
  • C30
  • D900

Q8:

For what value of 𝑘 is 𝑥 = 1 introduced as an extraneous solution to the equation 𝑥 = √ 𝑥 + 3 + 𝑘 when it is solved in the natural way?

Q9:

In order to solve the equation , Fady started with

What solutions did Fady find?

Q10:

Given that √ 7 + 1 8 9 = 5 + 𝑥 , find 𝑥 .

Q11:

What is the solution set of the equation 𝑥 + √ 𝑥 + 2 0 𝑥 + 1 0 0 = 5 ?

  • A 2 3
  • B 5 2
  • C − 2 3
  • D − 5 2

Q12:

Solve √ 𝑦 = 4 0 .

Q13:

Find the value of 𝑥 given that 3 √ 𝑥 = 1 0 .

  • A 100
  • B 10
  • C 1 0 3
  • D 1 000
  • E 30

Q14:

Solve √ 𝑥 = 2 √ 7 + 5 .

  • A 5 3 + 1 0 √ 7
  • B 5 3 − 2 0 √ 7
  • C 3 3 + 2 0 √ 7
  • D 5 3 + 2 0 √ 7
  • E 3 3 − 1 0 √ 7

Q15:

Find the solution set of √ 𝑥 − 9 = 4 1 2 in ℝ .

  • A { − 6 2 5 , 6 2 5 }
  • B { 2 5 }
  • C { − 2 5 , 2 5 }
  • D { 6 2 5 }

Q16:

Solve | | √ − 0 . 5 1 2 | | = √ 𝑥 3 .

Q17:

Solve √ 𝑦 = 8 . 5 .

Q18:

Consider the following argument for solving − 3 = √ 𝑥 :

  1. − 3 = √ 𝑥
  2. ( − 3 ) = √ 𝑥
  3. 9 = 𝑥 .

What mistake was made?

  • Anot checking for extraneous solutions
  • Bsquaring − 3 does not give 9

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