Worksheet: Solving Quadratics by Completing the Square

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In this worksheet, we will practice solving quadratics by completing the square and completing the square and then solving the equation using inverses.

Q1:

Solve the equation 𝑥 − 1 4 𝑥 + 3 8 = 0 2 by completing the square.

  • A 𝑥 = − 4 or 𝑥 = 1 8
  • B 𝑥 = 1 1 + √ 7 or 𝑥 = 1 1 − √ 7
  • C 𝑥 = 1 8 or 𝑥 = − 1 8
  • D 𝑥 = 7 + √ 1 1 or 𝑥 = 7 − √ 1 1
  • E 𝑥 = 7 or 𝑥 = 1 1

Q2:

By completing the square, solve the equation 𝑥 − 2 √ 3 𝑥 + 1 = 0 2 .

  • A 𝑥 = √ 3 6 , 𝑥 = − √ 3 6
  • B 𝑥 = 2 √ 3 , 𝑥 = − 2 √ 3
  • C 𝑥 = 1 , 𝑥 = − 1
  • D 𝑥 = √ 3 + √ 2 , 𝑥 = √ 3 − √ 2
  • E 𝑥 = − 1 + 2 √ 3 , 𝑥 = − − 1 + 2 √ 3

Q3:

By completing the square, solve the equation 𝑥 − 𝑥 = 3 4 2 .

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

Q4:

By completing the square, solve the equation 1 + 𝑥 = 𝑥 2 .

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

Q5:

By completing the square, solve the equation 𝑥 = 𝑥 + 3 4 2 .

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

Q6:

By completing the square, write all of the solutions to 𝑥 − 4 𝑥 + 1 = 0 2 .

  • A 2 − √ 3
  • B 2 + √ 3
  • C3
  • D 2 + √ 3 , 2 − √ 3
  • E5

Q7:

By completing the square, solve the equation 3 𝑥 − 2 𝑥 = 2 2 .

  • A 𝑥 = 1 3 + √ 5 3 𝑖 , 𝑥 = 1 3 − √ 5 3 𝑖
  • B 𝑥 = 1 3 + √ 7 3
  • C 𝑥 = − 1 3 + √ 7 3 , 𝑥 = − 1 3 − √ 7 3
  • D 𝑥 = 1 3 + √ 7 3 , 𝑥 = 1 3 − √ 7 3
  • E 𝑥 = 4 3 , 𝑥 = − 4 3

Q8:

By completing the square, write all of the solutions to 𝑥 + 4 𝑥 + 1 = 0 2 .

  • A − 2 − √ 3
  • B − 2 + √ 3
  • C − 2
  • D − 2 + √ 3 , − 2 − √ 3
  • E − 2 + √ 1 2 , − 2 − √ 1 2

Q9:

Solve the equation 𝑥 + 2 2 𝑥 + 1 1 4 = 0 2 by completing the square.

  • A 𝑥 = − 1 8 or 𝑥 = − 4
  • B 𝑥 = 7 + √ 1 1 or 𝑥 = 7 − √ 1 1
  • C 𝑥 = 4 or 𝑥 = − 4
  • D 𝑥 = − 1 1 + √ 7 or 𝑥 = − 1 1 − √ 7
  • E 𝑥 = − 1 1 or 𝑥 = 7

Q10:

Solve the equation 𝑥 − 1 4 𝑥 + 4 4 = 0 2 by completing the square.

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

Q11:

Solve the equation 𝑥 − 2 0 𝑥 + 7 7 = 0 2 by completing the square.

  • A 𝑥 = − 1 3 or 𝑥 = 3 3
  • B 𝑥 = 2 3 + √ 1 0 or 𝑥 = 2 3 − √ 1 0
  • C 𝑥 = 3 3 or 𝑥 = − 3 3
  • D 𝑥 = 1 0 + √ 2 3 or 𝑥 = 1 0 − √ 2 3
  • E 𝑥 = 1 0 or 𝑥 = 2 3

Q12:

Solve the equation 𝑥 − 4 𝑥 − 1 = 0 2 by completing the square.

  • A 𝑥 = − 3 or 𝑥 = 7
  • B 𝑥 = 5 + √ 2 or 𝑥 = 5 − √ 2
  • C 𝑥 = 7 or 𝑥 = − 7
  • D 𝑥 = 2 + √ 5 or 𝑥 = 2 − √ 5
  • E 𝑥 = 2 or 𝑥 = 5

Q13:

Solve the equation 𝑥 + 4 𝑥 + 1 = 0 2 by completing the square.

  • A 𝑥 = − 5 or 𝑥 = 1
  • B 𝑥 = 3 + √ 2 or 𝑥 = 3 − √ 2
  • C 𝑥 = 1 or 𝑥 = − 1
  • D 𝑥 = − 2 + √ 3 or 𝑥 = − 2 − √ 3
  • E 𝑥 = − 2 or 𝑥 = 3

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