Worksheet: Introduction to Completing the Square

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In this worksheet, we will practice completing the square for expressions where the coefficient of the leading term is one or otherwise.

Q1:

Given that 𝑥 − 1 0 𝑥 = ( 𝑥 + 𝑝 ) + 𝑞 2 2 , what are the values of 𝑝 and 𝑞 ?

  • A 𝑝 = 5 , 𝑞 = − 2 5
  • B 𝑝 = − 5 , 𝑞 = 2 5
  • C 𝑝 = − 1 0 , 𝑞 = − 1 0 0
  • D 𝑝 = − 5 , 𝑞 = − 2 5
  • E 𝑝 = 1 0 , 𝑞 = − 1 0 0

Q2:

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

  • A 𝑝 = 2 , 𝑞 = 1
  • B 𝑝 = 1 , 𝑞 = 5
  • C 𝑝 = 2 , 𝑞 = 5
  • D 𝑝 = 1 , 𝑞 = 4
  • E 𝑝 = 5 , 𝑞 = 1

Q3:

Given that 3 𝑥 + 3 𝑥 + 5 = 𝑎 ( 𝑥 + 𝑝 ) + 𝑞 2 2 , what are the values of 𝑎 , 𝑝 , and 𝑞 ?

  • A 𝑎 = 3 , 𝑝 = 1 2 , 𝑞 = 1 9 4
  • B 𝑎 = 3 , 𝑝 = 3 2 , 𝑞 = 1 1 4
  • C 𝑎 = 5 , 𝑝 = 3 1 0 , 𝑞 = 9 1 2 0
  • D 𝑎 = 3 , 𝑝 = 1 2 , 𝑞 = 1 7 4
  • E 𝑎 = 5 , 𝑝 = 3 2 , 𝑞 = 1 1 4

Q4:

What is the vertex form of the function 𝑓 ( 𝑥 ) = − 𝑥 + 6 𝑥 + 5 2 ?

  • A 𝑓 ( 𝑥 ) = − ( 𝑥 + 6 ) − 1 2
  • B 𝑓 ( 𝑥 ) = − ( 𝑥 + 3 ) + 1 4 2
  • C 𝑓 ( 𝑥 ) = − ( 𝑥 − 6 ) + 1 2
  • D 𝑓 ( 𝑥 ) = − ( 𝑥 − 3 ) + 1 4 2
  • E 𝑓 ( 𝑥 ) = ( 𝑥 − 3 ) + 1 4 2

Q5:

In completing the square for quadratic function 𝑓 ( 𝑥 ) = 𝑥 + 1 4 𝑥 + 4 6 2 , you arrive at the expression ( 𝑥 − 𝑏 ) + 𝑐 2 . What is the value of 𝑐 ?

Q6:

Given that − 𝑥 + 3 𝑥 + 4 = 𝑎 ( 𝑥 + 𝑝 ) + 𝑞 2 2 , what are the values of 𝑎 , 𝑝 , and 𝑞 ?

  • A 𝑎 = 1 , 𝑝 = 3 2 , 𝑞 = 9 4
  • B 𝑎 = − 1 , 𝑝 = 3 2 , 𝑞 = 9 4
  • C 𝑎 = − 1 , 𝑝 = 3 , 𝑞 = 4
  • D 𝑎 = − 1 , 𝑝 = − 3 2 , 𝑞 = 2 5 4
  • E 𝑎 = 1 , 𝑝 = − 3 , 𝑞 = − 4

Q7:

What is the vertex form of the function 𝑓 ( 𝑥 ) = 5 𝑥 − 𝑥 + 1 2 ?

  • A 𝑓 ( 𝑥 ) = 5 ( 𝑥 − 1 ) − 4 2
  • B 𝑓 ( 𝑥 ) = 5 𝑥 − 1 1 0 + 9 5 1 0 0 2
  • C 𝑓 ( 𝑥 ) = 5 ( 𝑥 + 1 ) − 4 2
  • D 𝑓 ( 𝑥 ) = 5 𝑥 − 1 1 0 + 9 5 1 0 0 2
  • E 𝑓 ( 𝑥 ) = 5 𝑥 − 1 1 0 + 1 5 1 0 0 2

Q8:

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

  • A 𝑥 + 1 3 2 = 3 0 2
  • B 𝑥 − 1 3 2 = 2 8 9 4 2
  • C 𝑥 + 1 6 9 4 = 2 8 9 4 2
  • D 𝑥 + 1 3 2 = 2 8 9 4 2
  • E 𝑥 + 1 6 9 4 = 3 0 2

Q9:

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

  • A 𝑥 − 𝑏 6 = 𝑏 − 1 2 𝑐 3 6
  • B 𝑥 + 𝑏 3 = 𝑏 − 3 𝑐 9
  • C 𝑥 + 𝑏 6 = − 𝑐 3
  • D 𝑥 + 𝑏 6 = 𝑏 − 1 2 𝑐 3 6
  • E 𝑥 − 𝑏 3 = 𝑏 − 3 𝑐 9

Q10:

Write the equation 𝑥 + 6 𝑥 − 3 = 0 2 in completed square form.

  • A ( 𝑥 + 3 ) − 9 = 0 2
  • B ( 𝑥 − 3 ) − 1 2 = 0 2
  • C ( 𝑥 − 3 ) − 9 = 0 2
  • D ( 𝑥 + 3 ) − 1 2 = 0 2
  • E ( 𝑥 + 6 ) − 3 9 = 0 2

Q11:

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

  • A 𝑥 + 1 2 = 5 4 2
  • B 𝑥 − 1 4 = 5 4 2
  • C 𝑥 − 1 2 = 1 2
  • D 𝑥 − 1 2 = 5 4 2
  • E 𝑥 − 1 4 = 1 2

Q12:

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

  • A 𝑥 + 1 2 = − 1 2
  • B 𝑥 + 1 4 = − 3 4 2
  • C 𝑥 − 1 2 = − 3 4 2
  • D 𝑥 + 1 2 = − 3 4 2
  • E 𝑥 + 1 4 = − 1 2

Q13:

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

  • A 𝑥 − 1 3 = 1 9 2
  • B 𝑥 − 1 3 = 0 2
  • C 𝑥 = 1 9 2
  • D 𝑥 = 1 3 2

Q14:

Write the equation 𝑥 − 𝑥 = 3 4 2 in the form ( 𝑥 − 𝑝 ) = 𝑞 2 .

  • A 𝑥 + 1 2 = 1 2
  • B 𝑥 − 1 4 = 1 2
  • C 𝑥 − 1 2 = 3 4 2
  • D 𝑥 − 1 2 = 1 2
  • E 𝑥 − 1 4 = 3 4 2

Q15:

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

  • A 𝑥 − √ 3 = − 2 2
  • B ( 𝑥 − 3 ) = 2 2
  • C 𝑥 − √ 3 = − 1 2
  • D 𝑥 − √ 3 = 2 2
  • E ( 𝑥 + 3 ) = 2 2

Q16:

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

  • A 𝑥 + 𝑏 6 = 1 2
  • B 𝑥 − 𝑏 6 = 𝑏 + 1 2 3 6 2 2
  • C 𝑥 + 𝑏 3 6 = 𝑏 + 1 2 3 6 2 2 2
  • D 𝑥 + 𝑏 6 = 𝑏 + 1 2 3 6 2 2
  • E 𝑥 − 𝑏 3 6 = 𝑏 + 1 2 3 6 2 2 2

Q17:

Write the equation 𝑥 + 𝑏 𝑥 + 𝑐 = 0 2 in the form ( 𝑥 − 𝑝 ) = 𝑞 2 .

  • A 𝑥 + 𝑏 2 = − 𝑐 2
  • B 𝑥 − 𝑏 2 = 𝑏 − 4 𝑐 4 2 2
  • C 𝑥 + 𝑏 4 = 𝑏 − 4 𝑐 4 2 2 2
  • D 𝑥 + 𝑏 2 = 𝑏 − 4 𝑐 4 2 2
  • E 𝑥 + 𝑏 2 = 𝑏 + 4 𝑐 4 2 2

Q18:

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

  • A 𝑥 + 𝑏 2 𝑎 = − 𝑐 𝑎
  • B 𝑥 − 𝑏 2 𝑎 = 𝑏 − 4 𝑎 𝑐 4 𝑎
  • C 𝑥 + 𝑏 2 𝑎 = 𝑏 − 𝑎 𝑐 𝑎
  • D 𝑥 + 𝑏 2 𝑎 = 𝑏 − 4 𝑎 𝑐 4 𝑎
  • E 𝑥 − 𝑏 2 𝑎 = − 𝑐 𝑎

Q19:

Which of the following equations can be transformed into the equation 2 𝑥 + 2 8 𝑥 + 6 = 0 2 by expanding, rearranging, and multiplying by a scalar?

  • A ( 𝑥 + 7 ) = − 3 2
  • B ( 𝑥 − 7 ) = 4 6 2
  • C ( 𝑥 + 4 9 ) = 4 6 2
  • D ( 𝑥 + 7 ) = 4 6 2
  • E ( 𝑥 + 4 9 ) = − 3 2

Q20:

Given that 𝑥 − 𝑥 − 𝑐 = 0 2 can be written in the form ( 𝑥 − 𝑝 ) = 3 2 , find the value of 𝑐 .

  • A − 1 3 4
  • B 1 3 4
  • C3
  • D 1 1 4
  • E − 1 1 4

Q21:

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

  • A 𝑎 = 1 , 𝑎 = 0
  • B 𝑎 = 0 , 𝑎 = 1 − √ 5 2 , 𝑎 = 1 + √ 5 2
  • C 𝑎 = 1 , 𝑎 = − 1
  • D 𝑎 = 1 , 𝑎 = 0 , 𝑎 = − 1
  • E 𝑎 = − 2 + √ 5 , 𝑎 = 0 , 𝑎 = − 2 + √ 5

Q22:

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

Q23:

Which of the following equations can be expanded and rearranged to 𝑥 + 1 = 8 𝑥 2 ?

  • A ( 𝑥 − 4 ) = − 1 5 2
  • B ( 𝑥 + 4 ) = 1 5 2
  • C ( 𝑥 + 4 ) = − 1 5 2
  • D ( 𝑥 − 4 ) = 1 5 2
  • E ( 𝑥 − 8 ) = 1 5 2

Q24:

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

  • A when 𝑎 > 0
  • B when 𝑎 < 0 or 𝑎 > 0
  • C when 𝑎 < 0
  • D when 0 < 𝑎 < 1
  • Ewhen 𝑎 < 1

Q25:

Factorise fully 2 4 𝑥 𝑦 + 3 𝑥 + 4 8 𝑦 .

  • A ( 𝑥 + 𝑦 )
  • B ( 𝑥 + 4 𝑦 + 3 )
  • C 3 ( 4 𝑥 + 𝑦 )
  • D 3 ( 𝑥 + 4 𝑦 )
  • E ( 4 𝑥 + 𝑦 )

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