Worksheet: Solving Quadratics by Taking Square Roots

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In this worksheet, we will practice solving quadratic equations with no linear term using the square root property.

Q1:

Find the solution set of the equation 3 𝑥 − 4 4 = 0 2 in ℝ , giving values to one decimal place.

  • A { 2 3 . 0 , − 2 3 . 0 }
  • B { 8 . 1 , − 8 . 1 }
  • C { − 1 4 . 7 , 1 4 . 7 }
  • D { 3 . 8 , − 3 . 8 }

Q2:

Find the solution set of the equation 4 𝑥 = 𝑥 9 .

  • A { 3 6 , − 3 6 }
  • B { 6 }
  • C { 3 6 }
  • D { 6 , − 6 }
  • E { 4 , 9 }

Q3:

Find the solution set of 2 𝑥 = 3 2 𝑥 3 in ℝ .

  • A { 1 6 , − 1 6 }
  • B { 4 , − 4 }
  • C { 4 }
  • D { 0 , 4 , − 4 }
  • E { 0 , 1 6 , − 1 6 }

Q4:

Find the solution set of the equation ( 𝑥 − 1 7 ) = 0 2 in ℝ .

  • A { 1 7 , − 1 7 }
  • B { − 1 7 }
  • C { 2 8 9 }
  • D { 1 7 }

Q5:

If 𝑥 − 5 , 8 = 9 5 , √ 𝑦 2 3 , find all the possible values of 𝑥 and 𝑦 .

  • A 𝑥 = 1 0 0 , 𝑦 = 5 1 2
  • B 𝑥 = 1 0 , 𝑦 = 8
  • C 𝑥 = 1 0 or 𝑥 = − 1 0 , 𝑦 = 8
  • D 𝑥 = 1 0 or 𝑥 = − 1 0 , 𝑦 = 5 1 2

Q6:

Solve 𝑥 = 2 2 .

  • A 𝑥 = 2 , 𝑥 = − 2
  • B 𝑥 = √ 2
  • C 𝑥 = 4
  • D 𝑥 = √ 2 , 𝑥 = − √ 2
  • E 𝑥 = 2

Q7:

Solve the equation 𝑥 + 1 = √ 3 2 .

  • A 𝑥 = − √ 3 − 1
  • B 𝑥 = √ 3 − 1
  • C 𝑥 = √ 3 , 𝑥 = − √ 3
  • D 𝑥 = √ 3 − 1 , 𝑥 = − √ 3 − 1
  • E 𝑥 = √ 3 + 1 , 𝑥 = − √ 3 + 1

Q8:

Solve 𝑥 = 5 1 1 2 .

  • A 𝑥 = 1 1 5 , 𝑥 = − 1 1 5
  • B 𝑥 = 5 1 1
  • C 𝑥 = 1 1 5
  • D 𝑥 = 5 1 1 , 𝑥 = − 5 1 1
  • E 𝑥 = √ 5 1 1 , 𝑥 = − √ 5 1 1

Q9:

Solve the equation 𝑥 = 1 6 2 .

  • A 𝑥 = − 4
  • B 𝑥 = 4
  • C 𝑥 = 8 , 𝑥 = − 8
  • D 𝑥 = 4 , 𝑥 = − 4
  • E 𝑥 = 8

Q10:

Solve ( 𝑥 + 6 ) = 4 2 .

  • A 𝑥 = − 5 or 𝑥 = − 7
  • B 𝑥 = 8 or 𝑥 = 4
  • C 𝑥 = 7 or 𝑥 = 5
  • D 𝑥 = − 4 or 𝑥 = − 8
  • E 𝑥 = 8 or 𝑥 = − 8

Q11:

Solve the equation 𝑥 + 2 = √ 3 2 .

  • A 𝑥 = 2 − √ 3 𝑖
  • B 𝑥 = √ 3 − 2 , 𝑥 = − √ 3 − 2
  • C 𝑥 = − 2 − √ 3 𝑖
  • D 𝑥 = 2 − √ 3 𝑖 , 𝑥 = − 2 − √ 3 𝑖
  • E 𝑥 = √ 3 − 2 2 , 𝑥 = − √ 3 − 2 2

Q12:

Solve the equation ( 𝑥 − 2 ) = 7 2 .

  • A 𝑥 = 2 − √ 7
  • B 𝑥 = 2 + √ 7
  • C 𝑥 = 3 , 𝑥 = − 3
  • D 𝑥 = 2 + √ 7 , 𝑥 = 2 − √ 7
  • E 𝑥 = 9

Q13:

Solve 𝑥 + 5 = 1 2 2 .

  • A 𝑥 = √ 1 2 − 5
  • B 𝑥 = √ 1 7 , 𝑥 = − √ 1 7
  • C 𝑥 = √ 7
  • D 𝑥 = √ 7 , 𝑥 = − √ 7
  • E 𝑥 = √ 1 7

Q14:

Determine the solution set of the equation 4 4 𝑥 + 9 = 0 2 given that 𝑥 ∈ ℝ .

  • A − 3 √ 1 1 2 2
  • B 3 √ 1 1 2 2
  • C − 3 √ 1 1 2 2 , 3 √ 1 1 2 2
  • D ∅

Q15:

Given that 𝑥 = √ 1 3 − √ 1 1 √ 1 3 + √ 1 1 2 , find the possible values of 𝑥 .

  • A √ 2
  • B 2 √ 6 , − 2 √ 6
  • C 2 √ 6
  • D √ 2 , − √ 2

Q16:

Find all the possible values of 𝑥 for which 2 𝑥 = 5 0 2 .

  • A √ 5 , − √ 5
  • B 25
  • C 10
  • D 5 , − 5

Q17:

Find all the values of 𝑥 which satisfy 𝑥 3 = 4 8 𝑥 .

  • A − 1 2
  • B12
  • C 1 4 4 , − 1 4 4
  • D 1 2 , − 1 2
  • E144

Q18:

Solve 𝑥 = 4 2 .

  • A 𝑥 = 2
  • B 𝑥 = 8 or 𝑥 = − 8
  • C 𝑥 = 1 6 or 𝑥 = − 1 6
  • D 𝑥 = 2 or 𝑥 = − 2
  • E 𝑥 = 8

Q19:

Solve 𝑥 + 5 = 1 4 .

  • A 𝑥 = 3
  • B 𝑥 = 9 or 𝑥 = − 9
  • C 𝑥 = 9
  • D 𝑥 = 3 or 𝑥 = − 3
  • E 𝑥 = 2

Q20:

Find the solution set of the equation ( 𝑥 − 5 ) = 1 0 0 2 in ℚ .

  • A { − 1 5 , 5 }
  • B ∅
  • C { 1 5 }
  • D { − 5 , 1 5 }

Q21:

Solve 𝑥 = 9 1 6 2 .

  • A 𝑥 = 3 1 6 or 𝑥 = − 3 1 6
  • B 𝑥 = 9 4 or 𝑥 = − 9 4
  • C 𝑥 = 3 4
  • D 𝑥 = 3 4 or 𝑥 = − 3 4
  • E 𝑥 = 3 1 6

Q22:

Given that 9 𝑥 + 2 = 4 2 , find the value of 𝑥 , and state whether it is a rational or irrational number.

  • A ± √ 6 3 , irrational number
  • B ± √ 2 3 , rational number
  • C ± √ 6 3 , rational number
  • D ± √ 2 3 , irrational number
  • E ± 2 9 , irrational number

Q23:

Write all of the values of 𝑥 which satisfy the equation 2 𝑥 + 8 = 1 3 . If necessary, give your answer(s) to 4 significant figures.

  • A1.257
  • B 1 2 . 5 7 , − 1 2 . 5 7
  • C1.5
  • D 1 . 2 5 7 , − 1 . 2 5 7
  • E − 1 . 2 5 7

Q24:

Find the solution set of the equation 𝑥 = 1 4 5 2 given that 𝑥 ∈ ℝ .

  • A √ 7 0 1 4 , − √ 7 0 1 4
  • B √ 7 0 5
  • C √ 7 0 1 4
  • D √ 7 0 5 , − √ 7 0 5
  • E √ 7 0 , − √ 7 0

Q25:

Solve 𝑥 = 0 . 3 6 2 .

  • A 𝑥 = 6 or 𝑥 = − 6
  • B 𝑥 = 0 . 0 6 or 𝑥 = − 0 . 0 6
  • C 𝑥 = 3 6 or 𝑥 = − 3 6
  • D 𝑥 = 0 . 6 or 𝑥 = − 0 . 6
  • E 𝑥 = 0 . 3 6 or 𝑥 = − 0 . 3 6

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