Worksheet: Solving Quadratics: Completing the Square‎

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


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

  • Aπ‘₯=βˆ’4 or π‘₯=18
  • Bπ‘₯=11+√7 or π‘₯=11βˆ’βˆš7
  • Cπ‘₯=7 or π‘₯=11
  • Dπ‘₯=7+√11 or π‘₯=7βˆ’βˆš11
  • Eπ‘₯=18 or π‘₯=βˆ’18


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

  • Aπ‘₯=ο„βˆ’1+2√3, π‘₯=βˆ’ο„βˆ’1+2√3
  • Bπ‘₯=1, π‘₯=βˆ’1
  • Cπ‘₯=√3+√2, π‘₯=√3βˆ’βˆš2
  • Dπ‘₯=2√3, π‘₯=βˆ’2√3
  • Eπ‘₯=√36, π‘₯=βˆ’βˆš36


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

  • Aπ‘₯=βˆ’12
  • Bπ‘₯=32, π‘₯=βˆ’12
  • Cπ‘₯=√32
  • Dπ‘₯=34, π‘₯=βˆ’12
  • Eπ‘₯=βˆ’32, π‘₯=12


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

  • Aπ‘₯=βˆšβˆ’5βˆ’12, π‘₯=√5βˆ’12
  • Bπ‘₯=12, π‘₯=βˆ’12
  • Cπ‘₯=√5+12
  • Dπ‘₯=√5+12, π‘₯=βˆ’βˆš5βˆ’12
  • Eπ‘₯=1, π‘₯=βˆ’1


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

  • Aπ‘₯=√32
  • Bπ‘₯=βˆ’12
  • Cπ‘₯=βˆ’32, π‘₯=12
  • Dπ‘₯=32, π‘₯=βˆ’12
  • Eπ‘₯=34, π‘₯=βˆ’12


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

  • A3
  • B2+√3, 2βˆ’βˆš3
  • C5
  • D2βˆ’βˆš3
  • E2+√3


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

  • Aπ‘₯=43, π‘₯=βˆ’43
  • Bπ‘₯=13+√73, π‘₯=13βˆ’βˆš73
  • Cπ‘₯=13+√53𝑖, π‘₯=13βˆ’βˆš53𝑖
  • Dπ‘₯=13+√73
  • Eπ‘₯=βˆ’13+√73, π‘₯=βˆ’13βˆ’βˆš73


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

  • Aβˆ’2
  • Bβˆ’2+√3,βˆ’2βˆ’βˆš3
  • Cβˆ’2+√3
  • Dβˆ’2+√12,βˆ’2βˆ’βˆš12
  • Eβˆ’2βˆ’βˆš3


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

  • Aπ‘₯=βˆ’18 or π‘₯=βˆ’4
  • Bπ‘₯=7+√11 or π‘₯=7βˆ’βˆš11
  • Cπ‘₯=βˆ’11 or π‘₯=7
  • Dπ‘₯=βˆ’11+√7 or π‘₯=βˆ’11βˆ’βˆš7
  • Eπ‘₯=4 or π‘₯=βˆ’4


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

  • Aπ‘₯=2 or π‘₯=12
  • Bπ‘₯=5+√7 or π‘₯=5βˆ’βˆš7
  • Cπ‘₯=7 or π‘₯=5
  • Dπ‘₯=7+√5 or π‘₯=7βˆ’βˆš5
  • Eπ‘₯=12 or π‘₯=βˆ’12


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

  • Aπ‘₯=βˆ’13 or π‘₯=33
  • Bπ‘₯=23+√10 or π‘₯=23βˆ’βˆš10
  • Cπ‘₯=10 or π‘₯=23
  • Dπ‘₯=10+√23 or π‘₯=10βˆ’βˆš23
  • Eπ‘₯=33 or π‘₯=βˆ’33


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

  • Aπ‘₯=βˆ’3 or π‘₯=7
  • Bπ‘₯=5+√2 or π‘₯=5βˆ’βˆš2
  • Cπ‘₯=2 or π‘₯=5
  • Dπ‘₯=2+√5 or π‘₯=2βˆ’βˆš5
  • Eπ‘₯=7 or π‘₯=βˆ’7


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

  • Aπ‘₯=βˆ’5 or π‘₯=1
  • Bπ‘₯=3+√2 or π‘₯=3βˆ’βˆš2
  • Cπ‘₯=βˆ’2 or π‘₯=3
  • Dπ‘₯=βˆ’2+√3 or π‘₯=βˆ’2βˆ’βˆš3
  • Eπ‘₯=1 or π‘₯=βˆ’1


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

  • Aπ‘₯=4+√26, π‘₯=4βˆ’βˆš26
  • Bπ‘₯=√264, π‘₯=βˆ’βˆš264
  • Cπ‘₯=βˆ’4+√26, π‘₯=βˆ’4βˆ’βˆš26.
  • Dπ‘₯=√26, π‘₯=βˆ’βˆš26
  • Eπ‘₯=4√26, π‘₯=βˆ’4√26


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?

  • A254
  • B814
  • C9
  • D92
  • Eβˆ’814


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’12=0
  • D(π‘₯+6)βˆ’39=0
  • E(π‘₯+3)βˆ’12=0


Solve π‘₯+2π‘₯βˆ’3=0.

  • Aπ‘₯=βˆ’1 or π‘₯=3
  • Bπ‘₯=βˆ’1
  • Cπ‘₯=1 or π‘₯=βˆ’3
  • Dπ‘₯=βˆ’1 or π‘₯=βˆ’3
  • Eπ‘₯=3


Solve π‘₯+3π‘₯βˆ’6=0.

  • Aπ‘₯=3+√332 or π‘₯=3βˆ’βˆš332
  • Bπ‘₯=βˆ’3+√33 or π‘₯=βˆ’3βˆ’βˆš33
  • Cπ‘₯=βˆ’3+√332 or π‘₯=3βˆ’βˆš332
  • Dπ‘₯=3+√33 or π‘₯=3βˆ’βˆš33
  • Eπ‘₯=βˆ’3+√332 or π‘₯=βˆ’3βˆ’βˆš332

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