Lesson Worksheet: Solving Quadratics: Completing the Square‎ Mathematics

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

Q1:

Solve the equation 𝑥−14𝑥+38=0 by completing the square.

  • A𝑥=−4 or 𝑥=18
  • B𝑥=11+√7 or 𝑥=11−√7
  • C𝑥=7 or 𝑥=11
  • D𝑥=7+√11 or 𝑥=7−√11
  • E𝑥=18 or 𝑥=−18

Q2:

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

  • A𝑥=−1+2√3, 𝑥=−−1+2√3
  • B𝑥=1, 𝑥=−1
  • C𝑥=√3+√2, 𝑥=√3−√2
  • D𝑥=2√3, 𝑥=−2√3
  • E𝑥=√36, 𝑥=−√36

Q3:

By completing the square, solve the equation 𝑥−𝑥=34.

  • A𝑥=−12
  • B𝑥=32, 𝑥=−12
  • C𝑥=√32
  • D𝑥=34, 𝑥=−12
  • E𝑥=−32, 𝑥=12

Q4:

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

  • A𝑥=√−5−12, 𝑥=√5−12
  • B𝑥=12, 𝑥=−12
  • C𝑥=√5+12
  • D𝑥=√5+12, 𝑥=−√5−12
  • E𝑥=1, 𝑥=−1

Q5:

By completing the square, solve the equation 𝑥=𝑥+34.

  • A𝑥=√32
  • B𝑥=−12
  • C𝑥=−32, 𝑥=12
  • D𝑥=32, 𝑥=−12
  • E𝑥=34, 𝑥=−12

Q6:

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

  • A𝑥=43, 𝑥=−43
  • B𝑥=13+√73, 𝑥=13−√73
  • C𝑥=13+√53𝑖, 𝑥=13−√53𝑖
  • D𝑥=13+√73
  • E𝑥=−13+√73, 𝑥=−13−√73

Q7:

Solve the equation 𝑥+22𝑥+114=0 by completing the square.

  • A𝑥=−18 or 𝑥=−4
  • B𝑥=7+√11 or 𝑥=7−√11
  • C𝑥=−11 or 𝑥=7
  • D𝑥=−11+√7 or 𝑥=−11−√7
  • E𝑥=4 or 𝑥=−4

Q8:

Solve the equation 𝑥−14𝑥+44=0 by completing the square.

  • A𝑥=2 or 𝑥=12
  • B𝑥=5+√7 or 𝑥=5−√7
  • C𝑥=7 or 𝑥=5
  • D𝑥=7+√5 or 𝑥=7−√5
  • E𝑥=12 or 𝑥=−12

Q9:

Solve the equation 𝑥−20𝑥+77=0 by completing the square.

  • A𝑥=−13 or 𝑥=33
  • B𝑥=23+√10 or 𝑥=23−√10
  • C𝑥=10 or 𝑥=23
  • D𝑥=10+√23 or 𝑥=10−√23
  • E𝑥=33 or 𝑥=−33

Q10:

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

  • A𝑥=−3 or 𝑥=7
  • B𝑥=5+√2 or 𝑥=5−√2
  • C𝑥=2 or 𝑥=5
  • D𝑥=2+√5 or 𝑥=2−√5
  • E𝑥=7 or 𝑥=−7

Q11:

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

  • A𝑥=−5 or 𝑥=1
  • B𝑥=3+√2 or 𝑥=3−√2
  • C𝑥=−2 or 𝑥=3
  • D𝑥=−2+√3 or 𝑥=−2−√3
  • E𝑥=1 or 𝑥=−1

Q12:

By completing the square, solve the equation 𝑥+8𝑥−10=0.

  • A𝑥=4+√26, 𝑥=4−√26
  • B𝑥=√264, 𝑥=−√264
  • C𝑥=−4+√26, 𝑥=−4−√26.
  • D𝑥=√26, 𝑥=−√26
  • E𝑥=4√26, 𝑥=−4√26

Q13:

While completing the square, James transformed 𝑥+9𝑥−5=0 into 𝑥+9𝑥=5. He then added a number to both sides. What number did he add?

  • A254
  • B814
  • C9
  • D92
  • E−814

Q14:

Write the equation 𝑥+6𝑥−3=0 in completed square form.

  • A(𝑥+3)−9=0
  • B(𝑥−3)−9=0
  • C(𝑥−3)−12=0
  • D(𝑥+6)−39=0
  • E(𝑥+3)−12=0

Q15:

Solve 𝑥+3𝑥−6=0.

  • A𝑥=3+√332 or 𝑥=3−√332
  • B𝑥=−3+√33 or 𝑥=−3−√33
  • C𝑥=−3+√332 or 𝑥=3−√332
  • D𝑥=3+√33 or 𝑥=3−√33
  • E𝑥=−3+√332 or 𝑥=−3−√332

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