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+9−2𝑥=3 introduces an extraneous solution. What is that extraneous solution?

  • A𝑥=117
  • B𝑥=3
  • C𝑥=−114
  • D𝑥=−2
  • E𝑥=−34

Q2:

Solve √𝑥−5=−1.

  • A𝑥=6
  • B𝑥=4
  • C𝑥=36
  • D𝑥=16

Q3:

Solve √𝑥+4+2=7.

  • A𝑥=21
  • B𝑥=1
  • C𝑥=29
  • D𝑥=9
  • E𝑥=43

Q4:

Find the value of 𝑥 given that √𝑥=−√36.

Q5:

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

  • A{−512}
  • B{−6}
  • C{−216}
  • D{216}
  • E{512}

Q6:

Find the value of 𝑥 for which √1,225=√400+√𝑥.

  • A𝑥=4
  • B𝑥=100
  • C𝑥=9
  • D𝑥=225
  • E𝑥=169

Q7:

Given that √𝑥+34=4, find the value of 𝑥.

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 𝑥+√𝑥=6, Michael started with 𝑥+√𝑥=6𝑥+√𝑥−6=0𝑥−6=−√𝑥(𝑥−6)=𝑥.

What solutions did Michael find?

Q10:

What is the solution set of the equation 𝑥+√𝑥+20𝑥+100=5?

  • A−23
  • B−52
  • C52
  • D23

Q11:

Solve √𝑦=40.

  • A𝑦=√40
  • B𝑦=80
  • C𝑦=20
  • D𝑦=40
  • E𝑦=1,600

Q12:

Find the value of 𝑥 given that √𝑥=10.

Q13:

Solve √𝑥=2√7+5.

  • A53−20√7
  • B53+20√7
  • C33−10√7
  • D53+10√7
  • E33+20√7

Q14:

Find the solution set of √𝑥−9=4 in ℝ.

  • A{−25,25}
  • B{−625,625}
  • C{25}
  • D{625}

Q15:

Solve ||√−0.512||=√𝑥.

Q16:

Solve √𝑦=8.5.

  • A𝑦=4.25
  • B𝑦=17
  • C𝑦=72.25
  • D𝑦=8.5
  • E𝑦=6.5

Q17:

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

  1. −3=√𝑥
  2. (−3)=√𝑥
  3. 9=𝑥.

What mistake was made?

  • Asquaring −3 does not give 9
  • Bnot checking for extraneous solutions

Q18:

Find the solution set of the equation (14𝑥−21)=√400 given that 𝑥∈ℝ.

  • A4114
  • B−114,−4114
  • C−27,3421
  • D4114,114
  • E4114,−4114

Q19:

Solve √676+𝑥=7.

Q20:

Given that √(5𝑥−7)(25𝑥−70𝑥+49)=1, find the value of 𝑥.

  • A−65
  • B125
  • C47
  • D85

Q21:

Write all values of 𝑥 which satisfy the equation √𝑥+1−𝑥=1.

  • A3
  • B0
  • C−1
  • D0,−1
  • E8

Q22:

Solve √𝑥−7=3.

  • A𝑥=12
  • B𝑥=16
  • C𝑥=2
  • D𝑥=10
  • E𝑥=4

Q23:

Write all the values of 𝑥 which satisfy the equation √2𝑥−1+𝑥=2.

Q24:

Daniel starts to solve 3+√𝑥+6=2𝑥 as follows: 3+√𝑥+6=2𝑥√𝑥+6=2𝑥−3𝑥+6=(2𝑥−3)𝑥+6=4𝑥−12𝑥+9.

Finish the solution for him.

  • A𝑥=14
  • B𝑥=13
  • C𝑥=3
  • D𝑥=2
  • E𝑥=12

Q25:

Squaring both sides of √𝑥+3=𝑥−3 introduces an extraneous solution. What is that extraneous solution?

  • A𝑥=1
  • B𝑥=−3
  • C𝑥=−1
  • D𝑥=4
  • E𝑥=3

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