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?

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

Q2:

Solve √𝑥−5=−1.

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

Q3:

Solve √𝑥+4+2=7.

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

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

Q6:

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

Q7:

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

  • A900
  • B30
  • C324
  • D27,000

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

Q11:

Solve √𝑦=40.

Q12:

Find the value of 𝑥 given that √𝑥=10.

  • A30
  • B100
  • C10
  • D1,000
  • E 1 0 3

Q13:

Solve √𝑥=2√7+5.

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

Q14:

Find the solution set of √𝑥−9=4 in ℝ.

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

Q15:

Solve ||√−0.512||=√𝑥.

Q16:

Solve √𝑦=8.5.

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.

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

Q19:

Solve √676+𝑥=7.

Q20:

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

  • A − 6 5
  • B 1 2 5
  • C 4 7
  • D 8 5

Q21:

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

  • A3
  • B0
  • C − 1
  • D 0 , − 1
  • E8

Q22:

Solve √𝑥−7=3.

  • A 𝑥 = 1 2
  • B 𝑥 = 1 6
  • C 𝑥 = 2
  • D 𝑥 = 1 0
  • E 𝑥 = 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.

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

Q25:

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

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

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