Lesson Worksheet: Completing the Square Mathematics

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

Q1:

Given that 𝑥10𝑥=(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)=(𝑥3)+14
  • C𝑓(𝑥)=(𝑥6)+1
  • D𝑓(𝑥)=(𝑥3)+14
  • E𝑓(𝑥)=(𝑥+3)+14

Q5:

In completing the square for quadratic function 𝑓(𝑥)=𝑥+14𝑥+46, you arrive at the expression (𝑥𝑏)+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)=5𝑥𝑥+1?

  • A𝑓(𝑥)=5𝑥110+95100
  • B𝑓(𝑥)=5𝑥110+95100
  • C𝑓(𝑥)=5(𝑥1)4
  • D𝑓(𝑥)=5(𝑥+1)4
  • E𝑓(𝑥)=5𝑥110+15100

Q8:

Write the equation 𝑥=3013𝑥 in the form (𝑥𝑝)=𝑞.

  • A𝑥+132=2894
  • B𝑥+1694=2894
  • C𝑥+132=30
  • D𝑥+1694=30
  • E𝑥132=2894

Q9:

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

  • A𝑥+𝑏3=𝑏3𝑐9
  • B𝑥+𝑏6=𝑏12𝑐36
  • C𝑥𝑏6=𝑏12𝑐36
  • D𝑥+𝑏6=𝑐3
  • E𝑥𝑏3=𝑏3𝑐9

Q10:

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

  • A𝑥+12=54
  • B𝑥12=54
  • C𝑥14=54
  • D𝑥14=1
  • E𝑥12=1

Q11:

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

  • A𝑥+14=34
  • B𝑥+12=1
  • C𝑥+12=34
  • D𝑥12=34
  • E𝑥+14=1

Q12:

Write the equation 3𝑥1=0 in the form (𝑥𝑝)=𝑞.

  • A𝑥=19
  • B𝑥13=19
  • C𝑥=13
  • D𝑥13=0

Q13:

Write the equation 𝑥𝑥=34 in the form (𝑥𝑝)=𝑞.

  • A𝑥12=34
  • B𝑥14=1
  • C𝑥14=34
  • D𝑥+12=1
  • E𝑥12=1

Q14:

Write the equation 𝑥23𝑥+1=0 in the form (𝑥𝑝)=𝑞.

  • A(𝑥3)=2
  • B𝑥3=2
  • C(𝑥+3)=2
  • D𝑥3=1
  • E𝑥3=2

Q15:

Write the equation 3𝑥+𝑏𝑥1=0 in the form (𝑥𝑝)=𝑞.

  • A𝑥+𝑏6=𝑏+1236
  • B𝑥𝑏6=𝑏+1236
  • C𝑥𝑏36=𝑏+1236
  • D𝑥+𝑏6=1
  • E𝑥+𝑏36=𝑏+1236

Q16:

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

  • A𝑥+𝑏2=𝑏+4𝑐4
  • B𝑥𝑏2=𝑏4𝑐4
  • C𝑥+𝑏2=𝑏4𝑐4
  • D𝑥+𝑏4=𝑏4𝑐4
  • E𝑥+𝑏2=𝑐

Q17:

Write the equation 𝑎𝑥+𝑏𝑥+𝑐=0, where 𝑎0, in the form (𝑥𝑝)=𝑞.

  • A𝑥+𝑏2𝑎=𝑏4𝑎𝑐4𝑎
  • B𝑥+𝑏2𝑎=𝑏𝑎𝑐𝑎
  • C𝑥𝑏2𝑎=𝑏4𝑎𝑐4𝑎
  • D𝑥+𝑏2𝑎=𝑐𝑎
  • E𝑥𝑏2𝑎=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(𝑥7)=46

Q19:

Given that 𝑥𝑥𝑐=0 can be written in the form (𝑥𝑝)=3, find the value of 𝑐.

  • A134
  • B114
  • C114
  • 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=152,𝑎=1+52

Q21:

Given that (3𝑥2𝑦)=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(𝑥8)=15
  • B(𝑥4)=15
  • C(𝑥4)=15
  • D(𝑥+4)=15
  • E(𝑥+4)=15

Q23:

By writing 𝑥+2𝑎𝑥+𝑎=0 in the form (𝑥𝑝)=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+9𝑧2𝑦9𝑧
  • B2𝑛2𝑦+9𝑧6𝑦𝑧
  • C2𝑛2𝑦6𝑦𝑧+9𝑧2𝑦+6𝑦𝑧+9𝑧
  • D2𝑛2𝑦+9𝑧
  • E2𝑛2𝑦18𝑦𝑧+9𝑧2𝑦+18𝑦𝑧+9𝑧

Q25:

Factorize fully 4𝑥+9+8𝑥.

  • A2𝑥2𝑥32𝑥+2𝑥3
  • B2𝑥4𝑥3
  • C2𝑥4𝑥+32𝑥+4𝑥+3
  • D2𝑥2𝑥32𝑥+2𝑥3
  • E2𝑥2𝑥+32𝑥+2𝑥+3

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