Worksheet: Completing the Square

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

Q1:

Given that π‘₯βˆ’10π‘₯=(π‘₯+𝑝)+π‘žοŠ¨οŠ¨, what are the values of 𝑝 and π‘ž?

  • A𝑝=βˆ’5, π‘ž=βˆ’25
  • B𝑝=βˆ’10, π‘ž=βˆ’100
  • C𝑝=5, π‘ž=βˆ’25
  • D𝑝=βˆ’5, π‘ž=25
  • E𝑝=10, π‘ž=βˆ’100

Q2:

Given that π‘₯+2π‘₯+5=(π‘₯+𝑝)+π‘žοŠ¨οŠ¨, what are the values of 𝑝 and π‘ž?

  • A𝑝=2, π‘ž=5
  • B𝑝=1, π‘ž=4
  • C𝑝=5, π‘ž=1
  • D𝑝=1, π‘ž=5
  • E𝑝=2, π‘ž=1

Q3:

Given that 3π‘₯+3π‘₯+5=π‘Ž(π‘₯+𝑝)+π‘žοŠ¨οŠ¨, what are the values of π‘Ž,𝑝, and π‘ž?

  • Aπ‘Ž=5, 𝑝=310, π‘ž=9120
  • Bπ‘Ž=5, 𝑝=32, π‘ž=114
  • Cπ‘Ž=3, 𝑝=32, π‘ž=114
  • Dπ‘Ž=3, 𝑝=12, π‘ž=194
  • Eπ‘Ž=3, 𝑝=12, π‘ž=174

Q4:

What is the vertex form of the function 𝑓(π‘₯)=βˆ’π‘₯+6π‘₯+5?

  • A𝑓(π‘₯)=βˆ’(π‘₯+6)βˆ’1
  • B𝑓(π‘₯)=(π‘₯βˆ’3)+14
  • C𝑓(π‘₯)=βˆ’(π‘₯βˆ’6)+1
  • D𝑓(π‘₯)=βˆ’(π‘₯βˆ’3)+14
  • E𝑓(π‘₯)=βˆ’(π‘₯+3)+14

Q5:

In completing the square for quadratic function 𝑓(π‘₯)=π‘₯+14π‘₯+46, you arrive at the expression (π‘₯βˆ’π‘)+π‘οŠ¨. What is the value of 𝑐?

Q6:

Given that βˆ’π‘₯+3π‘₯+4=π‘Ž(π‘₯+𝑝)+π‘žοŠ¨οŠ¨, what are the values of π‘Ž, 𝑝, and π‘ž?

  • Aπ‘Ž=1, 𝑝=32, π‘ž=94
  • Bπ‘Ž=βˆ’1, 𝑝=32, π‘ž=94
  • Cπ‘Ž=βˆ’1, 𝑝=3, π‘ž=4
  • Dπ‘Ž=βˆ’1, 𝑝=βˆ’32, π‘ž=254
  • Eπ‘Ž=1, 𝑝=βˆ’3, π‘ž=βˆ’4

Q7:

What is the vertex form of the function 𝑓(π‘₯)=5π‘₯βˆ’π‘₯+1?

  • A𝑓(π‘₯)=ο€Ό5π‘₯βˆ’110+95100
  • B𝑓(π‘₯)=5ο€Όπ‘₯βˆ’110+95100
  • C𝑓(π‘₯)=5(π‘₯βˆ’1)βˆ’4
  • D𝑓(π‘₯)=5(π‘₯+1)βˆ’4
  • E𝑓(π‘₯)=5ο€Όπ‘₯βˆ’110+15100

Q8:

Write the equation π‘₯=30βˆ’13π‘₯ in the form (π‘₯βˆ’π‘)=π‘žοŠ¨.

  • Aο€Όπ‘₯+132=2894
  • Bο€Όπ‘₯+1694=2894
  • Cο€Όπ‘₯+132=30
  • Dο€Όπ‘₯+1694=30
  • Eο€Όπ‘₯βˆ’132=2894

Q9:

Write the equation 3π‘₯+𝑏π‘₯+𝑐=0 in the form (π‘₯βˆ’π‘)=π‘žοŠ¨.

  • Aο€½π‘₯+𝑏3=π‘βˆ’3𝑐9
  • Bο€½π‘₯+𝑏6=π‘βˆ’12𝑐36
  • Cο€½π‘₯βˆ’π‘6=π‘βˆ’12𝑐36
  • Dο€½π‘₯+𝑏6=βˆ’π‘3
  • Eο€½π‘₯βˆ’π‘3=π‘βˆ’3𝑐9

Q10:

Write the equation 1+π‘₯=π‘₯ in the form (π‘₯βˆ’π‘)=π‘žοŠ¨.

  • Aο€Όπ‘₯+12=54
  • Bο€Όπ‘₯βˆ’12=54
  • Cο€Όπ‘₯βˆ’14=54
  • Dο€Όπ‘₯βˆ’14=1
  • Eο€Όπ‘₯βˆ’12=1

Q11:

Write the equation π‘₯+π‘₯+1=0 in the form (π‘₯βˆ’π‘)=π‘žοŠ¨.

  • Aο€Όπ‘₯+14=βˆ’34
  • Bο€Όπ‘₯+12=βˆ’1
  • Cο€Όπ‘₯+12=βˆ’34
  • Dο€Όπ‘₯βˆ’12=βˆ’34
  • Eο€Όπ‘₯+14=βˆ’1

Q12:

Write the equation 3π‘₯βˆ’1=0 in the form (π‘₯βˆ’π‘)=π‘žοŠ¨.

  • Aπ‘₯=19
  • Bο€Όπ‘₯βˆ’13=19
  • Cπ‘₯=13
  • Dο€Όπ‘₯βˆ’13=0

Q13:

Write the equation π‘₯βˆ’π‘₯=34 in the form (π‘₯βˆ’π‘)=π‘žοŠ¨.

  • Aο€Όπ‘₯βˆ’12=34
  • Bο€Όπ‘₯βˆ’14=1
  • Cο€Όπ‘₯βˆ’14=34
  • Dο€Όπ‘₯+12=1
  • Eο€Όπ‘₯βˆ’12=1

Q14:

Write the equation π‘₯βˆ’2√3π‘₯+1=0 in the form (π‘₯βˆ’π‘)=π‘žοŠ¨.

  • A(π‘₯βˆ’3)=2
  • Bο€»π‘₯βˆ’βˆš3=βˆ’2
  • C(π‘₯+3)=2
  • Dο€»π‘₯βˆ’βˆš3=βˆ’1
  • Eο€»π‘₯βˆ’βˆš3=2

Q15:

Write the equation 3π‘₯+𝑏π‘₯βˆ’1=0 in the form (π‘₯βˆ’π‘)=π‘žοŠ¨.

  • Aο€½π‘₯+𝑏6=𝑏+1236
  • Bο€½π‘₯βˆ’π‘6=𝑏+1236
  • Cο€Ύπ‘₯βˆ’π‘36=𝑏+1236
  • Dο€½π‘₯+𝑏6=1
  • Eο€Ύπ‘₯+𝑏36=𝑏+1236

Q16:

Write the equation π‘₯+𝑏π‘₯+𝑐=0 in the form (π‘₯βˆ’π‘)=π‘žοŠ¨.

  • Aο€½π‘₯+𝑏2=𝑏+4𝑐4
  • Bο€½π‘₯βˆ’π‘2=π‘βˆ’4𝑐4
  • Cο€½π‘₯+𝑏2=π‘βˆ’4𝑐4
  • Dο€Ύπ‘₯+𝑏4=π‘βˆ’4𝑐4
  • Eο€½π‘₯+𝑏2=βˆ’π‘οŠ¨

Q17:

Write the equation π‘Žπ‘₯+𝑏π‘₯+𝑐=0, where π‘Žβ‰ 0, in the form (π‘₯βˆ’π‘)=π‘žοŠ¨.

  • Aο€½π‘₯+𝑏2π‘Žο‰=π‘βˆ’4π‘Žπ‘4π‘ŽοŠ¨οŠ¨οŠ¨
  • Bο€½π‘₯+𝑏2π‘Žο‰=π‘βˆ’π‘Žπ‘π‘ŽοŠ¨οŠ¨οŠ¨
  • Cο€½π‘₯βˆ’π‘2π‘Žο‰=π‘βˆ’4π‘Žπ‘4π‘ŽοŠ¨οŠ¨οŠ¨
  • Dο€½π‘₯+𝑏2π‘Žο‰=βˆ’π‘π‘ŽοŠ¨
  • Eο€½π‘₯βˆ’π‘2π‘Žο‰=βˆ’π‘π‘ŽοŠ¨

Q18:

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

  • A(π‘₯+7)=βˆ’3
  • B(π‘₯+49)=βˆ’3
  • C(π‘₯+49)=46
  • D(π‘₯+7)=46
  • E(π‘₯βˆ’7)=46

Q19:

Given that π‘₯βˆ’π‘₯βˆ’π‘=0 can be written in the form (π‘₯βˆ’π‘)=3, find the value of 𝑐.

  • Aβˆ’134
  • B114
  • Cβˆ’114
  • D134
  • E3

Q20:

Find the values of π‘Ž for which the equation π‘₯+2π‘Žπ‘₯+π‘Ž+π‘Ž=π‘ŽοŠ¨οŠ¨οŠ© is satisfied by only one value of π‘₯.

  • Aπ‘Ž=βˆ’2+√5,π‘Ž=0,π‘Ž=βˆ’2+√5
  • Bπ‘Ž=1,π‘Ž=0
  • Cπ‘Ž=1,π‘Ž=βˆ’1
  • Dπ‘Ž=1,π‘Ž=0,π‘Ž=βˆ’1
  • Eπ‘Ž=0,π‘Ž=1βˆ’βˆš52,π‘Ž=1+√52

Q21:

Given that (3π‘₯βˆ’2𝑦)=6 and 9π‘₯+4𝑦=6, find the value of π‘₯𝑦.

Q22:

Which of the following equations can be expanded and rearranged to π‘₯+1=8π‘₯?

  • A(π‘₯βˆ’8)=15
  • B(π‘₯βˆ’4)=βˆ’15
  • C(π‘₯βˆ’4)=15
  • D(π‘₯+4)=15
  • E(π‘₯+4)=βˆ’15

Q23:

By writing π‘₯+2π‘Žπ‘₯+π‘Ž=0 in the form (π‘₯βˆ’π‘)=π‘žοŠ¨, determine when the equation has no real roots.

  • Awhen 0<π‘Ž<1
  • Bwhen π‘Ž<0 or π‘Ž>0
  • Cwhen π‘Ž<1
  • Dwhen π‘Ž<0
  • Ewhen π‘Ž>0

Q24:

Factorize fully 8𝑦𝑛+162𝑧𝑛οŠͺοŠͺ.

  • A2𝑛2𝑦+9𝑧2π‘¦βˆ’9π‘§ο…οŠ¨οŠ¨οŠ¨οŠ¨οŠ¨
  • B2𝑛2𝑦+9π‘§ο…βˆ’6π‘¦π‘§οŠ¨οŠ¨οŠ¨οŠ¨οŠͺοŠͺ
  • C2𝑛2π‘¦βˆ’6𝑦𝑧+9𝑧2𝑦+6𝑦𝑧+9π‘§ο…οŠ¨οŠ¨οŠ¨οŠ¨οŠ¨
  • D2𝑛2𝑦+9π‘§ο…οŠ¨οŠ¨οŠ¨οŠ¨
  • E2𝑛2π‘¦βˆ’18𝑦𝑧+9𝑧2𝑦+18𝑦𝑧+9π‘§ο…οŠ¨οŠ¨οŠ¨οŠ¨οŠ¨

Q25:

Factorize fully 4π‘₯+9+8π‘₯οŠͺ.

  • Aο€Ή2π‘₯βˆ’2π‘₯βˆ’32π‘₯+2π‘₯βˆ’3ο…οŠ¨οŠ¨
  • Bο€Ή2π‘₯βˆ’4π‘₯βˆ’3ο…οŠ¨οŠ¨
  • Cο€Ή2π‘₯βˆ’4π‘₯+32π‘₯+4π‘₯+3ο…οŠ¨οŠ¨
  • Dο€Ή2π‘₯βˆ’2π‘₯βˆ’32π‘₯+2π‘₯βˆ’3ο…οŠ¨οŠ¨οŠ¨οŠ¨
  • Eο€Ή2π‘₯βˆ’2π‘₯+32π‘₯+2π‘₯+3ο…οŠ¨οŠ¨

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.