Lesson Worksheet: Completing the Square Mathematics

In this worksheet, we will practice completing the square to factor quadratic expressions.

Q1:

Given that 𝑥−10𝑥=(𝑥+𝑝)+𝑞, what are the values of 𝑝 and 𝑞?

  • A𝑝=−5, 𝑞=−25
  • B𝑝=−10, 𝑞=−100
  • C𝑝=5, 𝑞=−25
  • D𝑝=−5, 𝑞=25
  • E𝑝=10, 𝑞=−100

Q2:

Given that 𝑥+2𝑥+5=(𝑥+𝑝)+𝑞, what are the values of 𝑝 and 𝑞?

  • A𝑝=2, 𝑞=5
  • B𝑝=1, 𝑞=4
  • C𝑝=5, 𝑞=1
  • D𝑝=1, 𝑞=5
  • E𝑝=2, 𝑞=1

Q3:

Given that 3𝑥+3𝑥+5=𝑎(𝑥+𝑝)+𝑞, what are the values of 𝑎,𝑝, and 𝑞?

  • A𝑎=5, 𝑝=310, 𝑞=9120
  • B𝑎=5, 𝑝=32, 𝑞=114
  • C𝑎=3, 𝑝=32, 𝑞=114
  • D𝑎=3, 𝑝=12, 𝑞=194
  • E𝑎=3, 𝑝=12, 𝑞=174

Q4:

What is the vertex form of the function 𝑓(𝑥)=−𝑥+6𝑥+5?

  • A𝑓(𝑥)=−(𝑥+6)−1
  • B𝑓(𝑥)=(𝑥−3)+14
  • C𝑓(𝑥)=−(𝑥−6)+1
  • D𝑓(𝑥)=−(𝑥−3)+14
  • E𝑓(𝑥)=−(𝑥+3)+14

Q5:

In completing the square for quadratic function 𝑓(𝑥)=𝑥+14𝑥+46, you arrive at the expression (𝑥−𝑏)+𝑐. What is the value of 𝑐?

Q6:

Given that −𝑥+3𝑥+4=𝑎(𝑥+𝑝)+𝑞, what are the values of 𝑎, 𝑝, and 𝑞?

  • A𝑎=1, 𝑝=32, 𝑞=94
  • B𝑎=−1, 𝑝=32, 𝑞=94
  • C𝑎=−1, 𝑝=3, 𝑞=4
  • D𝑎=−1, 𝑝=−32, 𝑞=254
  • E𝑎=1, 𝑝=−3, 𝑞=−4

Q7:

What is the vertex form of the function 𝑓(𝑥)=5𝑥−𝑥+1?

  • A𝑓(𝑥)=5𝑥−110+95100
  • B𝑓(𝑥)=5𝑥−110+95100
  • C𝑓(𝑥)=5(𝑥−1)−4
  • D𝑓(𝑥)=5(𝑥+1)−4
  • E𝑓(𝑥)=5𝑥−110+15100

Q8:

Write the equation 𝑥=30−13𝑥 in the form (𝑥−𝑝)=𝑞.

  • A𝑥+132=2894
  • B𝑥+1694=2894
  • C𝑥+132=30
  • D𝑥+1694=30
  • E𝑥−132=2894

Q9:

Write the equation 3𝑥+𝑏𝑥+𝑐=0 in the form (𝑥−𝑝)=𝑞.

  • A𝑥+𝑏3=𝑏−3𝑐9
  • B𝑥+𝑏6=𝑏−12𝑐36
  • C𝑥−𝑏6=𝑏−12𝑐36
  • D𝑥+𝑏6=−𝑐3
  • E𝑥−𝑏3=𝑏−3𝑐9

Q10:

Write the equation 1+𝑥=𝑥 in the form (𝑥−𝑝)=𝑞.

  • A𝑥+12=54
  • B𝑥−12=54
  • C𝑥−14=54
  • D𝑥−14=1
  • E𝑥−12=1

Q11:

Write the equation 𝑥+𝑥+1=0 in the form (𝑥−𝑝)=𝑞.

  • A𝑥+14=−34
  • B𝑥+12=−1
  • C𝑥+12=−34
  • D𝑥−12=−34
  • E𝑥+14=−1

Q12:

Write the equation 3𝑥−1=0 in the form (𝑥−𝑝)=𝑞.

  • A𝑥=19
  • B𝑥−13=19
  • C𝑥=13
  • D𝑥−13=0

Q13:

Write the equation 𝑥−𝑥=34 in the form (𝑥−𝑝)=𝑞.

  • A𝑥−12=34
  • B𝑥−14=1
  • C𝑥−14=34
  • D𝑥+12=1
  • E𝑥−12=1

Q14:

Write the equation 𝑥−2√3𝑥+1=0 in the form (𝑥−𝑝)=𝑞.

  • A(𝑥−3)=2
  • B𝑥−√3=−2
  • C(𝑥+3)=2
  • D𝑥−√3=−1
  • E𝑥−√3=2

Q15:

Write the equation 3𝑥+𝑏𝑥−1=0 in the form (𝑥−𝑝)=𝑞.

  • A𝑥+𝑏6=𝑏+1236
  • B𝑥−𝑏6=𝑏+1236
  • C𝑥−𝑏36=𝑏+1236
  • D𝑥+𝑏6=1
  • E𝑥+𝑏36=𝑏+1236

Q16:

Write the equation 𝑥+𝑏𝑥+𝑐=0 in the form (𝑥−𝑝)=𝑞.

  • A𝑥+𝑏2=𝑏+4𝑐4
  • B𝑥−𝑏2=𝑏−4𝑐4
  • C𝑥+𝑏2=𝑏−4𝑐4
  • D𝑥+𝑏4=𝑏−4𝑐4
  • E𝑥+𝑏2=−𝑐

Q17:

Write the equation 𝑎𝑥+𝑏𝑥+𝑐=0, where 𝑎≠0, in the form (𝑥−𝑝)=𝑞.

  • A𝑥+𝑏2𝑎=𝑏−4𝑎𝑐4𝑎
  • B𝑥+𝑏2𝑎=𝑏−𝑎𝑐𝑎
  • C𝑥−𝑏2𝑎=𝑏−4𝑎𝑐4𝑎
  • D𝑥+𝑏2𝑎=−𝑐𝑎
  • E𝑥−𝑏2𝑎=−𝑐𝑎

Q18:

Which of the following equations can be transformed into the equation 2𝑥+28𝑥+6=0 by expanding, rearranging, and multiplying by a scalar?

  • A(𝑥+7)=−3
  • B(𝑥+49)=−3
  • C(𝑥+49)=46
  • D(𝑥+7)=46
  • E(𝑥−7)=46

Q19:

Given that 𝑥−𝑥−𝑐=0 can be written in the form (𝑥−𝑝)=3, find the value of 𝑐.

  • A−134
  • B114
  • C−114
  • D134
  • E3

Q20:

Find the values of 𝑎 for which the equation 𝑥+2𝑎𝑥+𝑎+𝑎=𝑎 is satisfied by only one value of 𝑥.

  • A𝑎=−2+√5,𝑎=0,𝑎=−2+√5
  • B𝑎=1,𝑎=0
  • C𝑎=1,𝑎=−1
  • D𝑎=1,𝑎=0,𝑎=−1
  • E𝑎=0,𝑎=1−√52,𝑎=1+√52

Q21:

Given that (3𝑥−2𝑦)=6 and 9𝑥+4𝑦=6, find the value of 𝑥𝑦.

Q22:

Which of the following equations can be expanded and rearranged to 𝑥+1=8𝑥?

  • A(𝑥−8)=15
  • B(𝑥−4)=−15
  • C(𝑥−4)=15
  • D(𝑥+4)=15
  • E(𝑥+4)=−15

Q23:

By writing 𝑥+2𝑎𝑥+𝑎=0 in the form (𝑥−𝑝)=𝑞, determine when the equation has no real roots.

  • Awhen 0<𝑎<1
  • Bwhen 𝑎<0 or 𝑎>0
  • Cwhen 𝑎<1
  • Dwhen 𝑎<0
  • Ewhen 𝑎>0

Q24:

Factorize fully 8𝑦𝑛+162𝑧𝑛.

  • A2𝑛2𝑦+9𝑧2𝑦−9𝑧
  • B2𝑛2𝑦+9𝑧−6𝑦𝑧
  • C2𝑛2𝑦−6𝑦𝑧+9𝑧2𝑦+6𝑦𝑧+9𝑧
  • D2𝑛2𝑦+9𝑧
  • E2𝑛2𝑦−18𝑦𝑧+9𝑧2𝑦+18𝑦𝑧+9𝑧

Q25:

Factorize fully 4𝑥+9+8𝑥.

  • A2𝑥−2𝑥−32𝑥+2𝑥−3
  • B2𝑥−4𝑥−3
  • C2𝑥−4𝑥+32𝑥+4𝑥+3
  • D2𝑥−2𝑥−32𝑥+2𝑥−3
  • E2𝑥−2𝑥+32𝑥+2𝑥+3

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