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Worksheet: Solving Quadratics by Completing the Square

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

Q1:

What are the zeros of the function 𝑓 ( π‘₯ ) = 2 ( π‘₯ βˆ’ 1 ) βˆ’ 7 2 ?

  • A βˆ’ 1 + ο„ž 7 2 and βˆ’ 1 βˆ’ ο„ž 7 2
  • B 1 + √ 7 2 and 1 βˆ’ √ 7 2
  • C βˆ’ 1 + √ 7 2 and βˆ’ 1 βˆ’ √ 7 2
  • D 1 + ο„ž 7 2 and 1 βˆ’ ο„ž 7 2
  • E 1 βˆ’ √ 7 2 and βˆ’ 1 βˆ’ √ 7 2

Q2:

Solve the equation π‘₯ βˆ’ 1 4 π‘₯ + 3 8 = 0 2 by completing the square.

  • A π‘₯ = βˆ’ 4 or π‘₯ = 1 8
  • B π‘₯ = 1 1 + √ 7 or π‘₯ = 1 1 βˆ’ √ 7
  • C π‘₯ = 1 8 or π‘₯ = βˆ’ 1 8
  • D π‘₯ = 7 + √ 1 1 or π‘₯ = 7 βˆ’ √ 1 1
  • E π‘₯ = 7 or π‘₯ = 1 1

Q3:

Solve the equation π‘₯ + 2 2 π‘₯ + 1 1 4 = 0 2 by completing the square.

  • A π‘₯ = βˆ’ 1 8 or π‘₯ = βˆ’ 4
  • B π‘₯ = 7 + √ 1 1 or π‘₯ = 7 βˆ’ √ 1 1
  • C π‘₯ = 4 or π‘₯ = βˆ’ 4
  • D π‘₯ = βˆ’ 1 1 + √ 7 or π‘₯ = βˆ’ 1 1 βˆ’ √ 7
  • E π‘₯ = βˆ’ 1 1 or π‘₯ = 7

Q4:

Solve the equation π‘₯ βˆ’ 1 4 π‘₯ + 4 4 = 0 2 by completing the square.

  • A π‘₯ = 2 or π‘₯ = 1 2
  • B π‘₯ = 5 + √ 7 or π‘₯ = 5 βˆ’ √ 7
  • C π‘₯ = 1 2 or π‘₯ = βˆ’ 1 2
  • D π‘₯ = 7 + √ 5 or π‘₯ = 7 βˆ’ √ 5
  • E π‘₯ = 7 or π‘₯ = 5

Q5:

Solve the equation π‘₯ βˆ’ 2 0 π‘₯ + 7 7 = 0 2 by completing the square.

  • A π‘₯ = βˆ’ 1 3 or π‘₯ = 3 3
  • B π‘₯ = 2 3 + √ 1 0 or π‘₯ = 2 3 βˆ’ √ 1 0
  • C π‘₯ = 3 3 or π‘₯ = βˆ’ 3 3
  • D π‘₯ = 1 0 + √ 2 3 or π‘₯ = 1 0 βˆ’ √ 2 3
  • E π‘₯ = 1 0 or π‘₯ = 2 3

Q6:

Solve the equation π‘₯ βˆ’ 4 π‘₯ βˆ’ 1 = 0 2 by completing the square.

  • A π‘₯ = βˆ’ 3 or π‘₯ = 7
  • B π‘₯ = 5 + √ 2 or π‘₯ = 5 βˆ’ √ 2
  • C π‘₯ = 7 or π‘₯ = βˆ’ 7
  • D π‘₯ = 2 + √ 5 or π‘₯ = 2 βˆ’ √ 5
  • E π‘₯ = 2 or π‘₯ = 5

Q7:

Solve the equation π‘₯ + 4 π‘₯ + 1 = 0 2 by completing the square.

  • A π‘₯ = βˆ’ 5 or π‘₯ = 1
  • B π‘₯ = 3 + √ 2 or π‘₯ = 3 βˆ’ √ 2
  • C π‘₯ = 1 or π‘₯ = βˆ’ 1
  • D π‘₯ = βˆ’ 2 + √ 3 or π‘₯ = βˆ’ 2 βˆ’ √ 3
  • E π‘₯ = βˆ’ 2 or π‘₯ = 3

Q8:

By completing the square, solve the equation π‘₯ βˆ’ 2 √ 3 π‘₯ + 1 = 0 2 .

  • A π‘₯ = √ 3 6 , π‘₯ = βˆ’ √ 3 6
  • B π‘₯ = 2 √ 3 , π‘₯ = βˆ’ 2 √ 3
  • C π‘₯ = 1 , π‘₯ = βˆ’ 1
  • D π‘₯ = √ 3 + √ 2 , π‘₯ = √ 3 βˆ’ √ 2
  • E π‘₯ =  βˆ’ 1 + 2 √ 3 , π‘₯ = βˆ’  βˆ’ 1 + 2 √ 3

Q9:

By completing the square, solve the equation 1 + π‘₯ = π‘₯ 2 .

  • A π‘₯ = √ βˆ’ 5 βˆ’ 1 2 , π‘₯ = √ 5 βˆ’ 1 2
  • B π‘₯ = √ 5 + 1 2
  • C π‘₯ = 1 , π‘₯ = βˆ’ 1
  • D π‘₯ = √ 5 + 1 2 , π‘₯ = βˆ’ √ 5 βˆ’ 1 2
  • E π‘₯ = 1 2 , π‘₯ = βˆ’ 1 2

Q10:

By completing the square, solve the equation π‘₯ = π‘₯ + 3 4 2 .

  • A π‘₯ = βˆ’ 3 2 , π‘₯ = 1 2
  • B π‘₯ = √ 3 2
  • C π‘₯ = 3 4 , π‘₯ = βˆ’ 1 2
  • D π‘₯ = 3 2 , π‘₯ = βˆ’ 1 2
  • E π‘₯ = βˆ’ 1 2

Q11:

By completing the square, solve the equation π‘₯ βˆ’ π‘₯ = 3 4 2 .

  • A π‘₯ = βˆ’ 3 2 , π‘₯ = 1 2
  • B π‘₯ = √ 3 2
  • C π‘₯ = 3 4 , π‘₯ = βˆ’ 1 2
  • D π‘₯ = 3 2 , π‘₯ = βˆ’ 1 2
  • E π‘₯ = βˆ’ 1 2

Q12:

By completing the square, write all of the solutions to π‘₯ βˆ’ 4 π‘₯ + 1 = 0 2 .

  • A 2 βˆ’ √ 3
  • B 2 + √ 3
  • C3
  • D 2 + √ 3 , 2 βˆ’ √ 3
  • E5

Q13:

By completing the square, solve the equation 3 π‘₯ βˆ’ 2 π‘₯ = 2 2 .

  • A π‘₯ = 1 3 + √ 5 3 𝑖 , π‘₯ = 1 3 βˆ’ √ 5 3 𝑖
  • B π‘₯ = 1 3 + √ 7 3
  • C π‘₯ = βˆ’ 1 3 + √ 7 3 , π‘₯ = βˆ’ 1 3 βˆ’ √ 7 3
  • D π‘₯ = 1 3 + √ 7 3 , π‘₯ = 1 3 βˆ’ √ 7 3
  • E π‘₯ = 4 3 , π‘₯ = βˆ’ 4 3

Q14:

By completing the square, write all of the solutions to π‘₯ + 4 π‘₯ + 1 = 0 2 .

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

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