Worksheet: Solving Quadratics: Completing the Square‎

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In this worksheet, we will practice solving quadratics by completing the square.

Q1:

Solve the equation π‘₯βˆ’14π‘₯+38=0 by completing the square.

  • A π‘₯ = βˆ’ 4 or π‘₯=18
  • B π‘₯ = 1 1 + √ 7 or π‘₯=11βˆ’βˆš7
  • C π‘₯ = 7 or π‘₯=11
  • D π‘₯ = 7 + √ 1 1 or π‘₯=7βˆ’βˆš11
  • E π‘₯ = 1 8 or π‘₯=βˆ’18

Q2:

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

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

Q3:

By completing the square, solve the equation π‘₯βˆ’π‘₯=34.

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

Q4:

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

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

Q5:

By completing the square, solve the equation π‘₯=π‘₯+34.

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

Q6:

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

  • A3
  • B 2 + √ 3 , 2 βˆ’ √ 3
  • C5
  • D 2 βˆ’ √ 3
  • E 2 + √ 3

Q7:

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

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

Q8:

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

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

Q9:

Solve the equation π‘₯+22π‘₯+114=0 by completing the square.

  • A π‘₯ = βˆ’ 1 8 or π‘₯=βˆ’4
  • B π‘₯ = 7 + √ 1 1 or π‘₯=7βˆ’βˆš11
  • C π‘₯ = βˆ’ 1 1 or π‘₯=7
  • D π‘₯ = βˆ’ 1 1 + √ 7 or π‘₯=βˆ’11βˆ’βˆš7
  • E π‘₯ = 4 or π‘₯=βˆ’4

Q10:

Solve the equation π‘₯βˆ’14π‘₯+44=0 by completing the square.

  • A π‘₯ = 2 or π‘₯=12
  • B π‘₯ = 5 + √ 7 or π‘₯=5βˆ’βˆš7
  • C π‘₯ = 7 or π‘₯=5
  • D π‘₯ = 7 + √ 5 or π‘₯=7βˆ’βˆš5
  • E π‘₯ = 1 2 or π‘₯=βˆ’12

Q11:

Solve the equation π‘₯βˆ’20π‘₯+77=0 by completing the square.

  • A π‘₯ = βˆ’ 1 3 or π‘₯=33
  • B π‘₯ = 2 3 + √ 1 0 or π‘₯=23βˆ’βˆš10
  • C π‘₯ = 1 0 or π‘₯=23
  • D π‘₯ = 1 0 + √ 2 3 or π‘₯=10βˆ’βˆš23
  • E π‘₯ = 3 3 or π‘₯=βˆ’33

Q12:

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

  • A π‘₯ = βˆ’ 3 or π‘₯=7
  • B π‘₯ = 5 + √ 2 or π‘₯=5βˆ’βˆš2
  • C π‘₯ = 2 or π‘₯=5
  • D π‘₯ = 2 + √ 5 or π‘₯=2βˆ’βˆš5
  • E π‘₯ = 7 or π‘₯=βˆ’7

Q13:

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

  • A π‘₯ = βˆ’ 5 or π‘₯=1
  • B π‘₯ = 3 + √ 2 or π‘₯=3βˆ’βˆš2
  • C π‘₯ = βˆ’ 2 or π‘₯=3
  • D π‘₯ = βˆ’ 2 + √ 3 or π‘₯=βˆ’2βˆ’βˆš3
  • E π‘₯ = 1 or π‘₯=βˆ’1

Q14:

By completing the square, solve the equation π‘₯+8π‘₯βˆ’10=0.

  • A π‘₯ = 4 + √ 2 6 , π‘₯ = 4 βˆ’ √ 2 6
  • B π‘₯ = √ 2 6 4 , π‘₯ = βˆ’ √ 2 6 4
  • C π‘₯ = βˆ’ 4 + √ 2 6 , π‘₯ = βˆ’ 4 βˆ’ √ 2 6 .
  • D π‘₯ = √ 2 6 , π‘₯ = βˆ’ √ 2 6
  • E π‘₯ = 4 √ 2 6 , π‘₯ = βˆ’ 4 √ 2 6

Q15:

While completing the square, James transformed π‘₯+9π‘₯βˆ’5=0 into π‘₯+9π‘₯=5. He then added a number to both sides. What number did he add?

  • A 2 5 4
  • B 8 1 4
  • C9
  • D 9 2
  • E βˆ’ 8 1 4

Q16:

Write the equation π‘₯+6π‘₯βˆ’3=0 in completed square form.

  • A ( π‘₯ + 3 ) βˆ’ 9 = 0 
  • B ( π‘₯ βˆ’ 3 ) βˆ’ 9 = 0 
  • C ( π‘₯ βˆ’ 3 ) βˆ’ 1 2 = 0 
  • D ( π‘₯ + 6 ) βˆ’ 3 9 = 0 
  • E ( π‘₯ + 3 ) βˆ’ 1 2 = 0 

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