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Worksheet: Completing the Square

Q1:

Given that π‘₯ βˆ’ 1 0 π‘₯ = ( π‘₯ + 𝑝 ) + π‘ž 2 2 , what are the values of 𝑝 and π‘ž ?

  • A 𝑝 = 5 , π‘ž = βˆ’ 2 5
  • B 𝑝 = βˆ’ 5 , π‘ž = 2 5
  • C 𝑝 = βˆ’ 1 0 , π‘ž = βˆ’ 1 0 0
  • D 𝑝 = βˆ’ 5 , π‘ž = βˆ’ 2 5
  • E 𝑝 = 1 0 , π‘ž = βˆ’ 1 0 0

Q2:

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

  • A 𝑝 = 2 , π‘ž = 1
  • B 𝑝 = 1 , π‘ž = 5
  • C 𝑝 = 2 , π‘ž = 5
  • D 𝑝 = 1 , π‘ž = 4
  • E 𝑝 = 5 , π‘ž = 1

Q3:

Given that 3 π‘₯ + 3 π‘₯ + 5 = π‘Ž ( π‘₯ + 𝑝 ) + π‘ž 2 2 , what are the values of π‘Ž , 𝑝 , and π‘ž ?

  • A π‘Ž = 3 , 𝑝 = 1 2 , π‘ž = 1 9 4
  • B π‘Ž = 3 , 𝑝 = 3 2 , π‘ž = 1 1 4
  • C π‘Ž = 5 , 𝑝 = 3 1 0 , π‘ž = 9 1 2 0
  • D π‘Ž = 3 , 𝑝 = 1 2 , π‘ž = 1 7 4
  • E π‘Ž = 5 , 𝑝 = 3 2 , π‘ž = 1 1 4

Q4:

Given that βˆ’ π‘₯ + 3 π‘₯ + 4 = π‘Ž ( π‘₯ + 𝑝 ) + π‘ž 2 2 , what are the values of π‘Ž , 𝑝 , and π‘ž ?

  • A π‘Ž = 1 , 𝑝 = 3 2 , π‘ž = 9 4
  • B π‘Ž = βˆ’ 1 , 𝑝 = 3 2 , π‘ž = 9 4
  • C π‘Ž = βˆ’ 1 , 𝑝 = 3 , π‘ž = 4
  • D π‘Ž = βˆ’ 1 , 𝑝 = βˆ’ 3 2 , π‘ž = 2 5 4
  • E π‘Ž = 1 , 𝑝 = βˆ’ 3 , π‘ž = βˆ’ 4

Q5:

What is the vertex form of the function 𝑓 ( π‘₯ ) = 5 π‘₯ βˆ’ π‘₯ + 1 2 ?

  • A 𝑓 ( π‘₯ ) = 5 ( π‘₯ βˆ’ 1 ) βˆ’ 4 2
  • B 𝑓 ( π‘₯ ) = ο€Ό 5 π‘₯ βˆ’ 1 1 0  + 9 5 1 0 0 2
  • C 𝑓 ( π‘₯ ) = 5 ( π‘₯ + 1 ) βˆ’ 4 2
  • D 𝑓 ( π‘₯ ) = 5 ο€Ό π‘₯ βˆ’ 1 1 0  + 9 5 1 0 0 2
  • E 𝑓 ( π‘₯ ) = 5 ο€Ό π‘₯ βˆ’ 1 1 0  + 1 5 1 0 0 2

Q6:

What is the vertex form of the function 𝑓 ( π‘₯ ) = βˆ’ π‘₯ + 6 π‘₯ + 5 2 ?

  • A 𝑓 ( π‘₯ ) = βˆ’ ( π‘₯ + 6 ) βˆ’ 1 2
  • B 𝑓 ( π‘₯ ) = βˆ’ ( π‘₯ + 3 ) + 1 4 2
  • C 𝑓 ( π‘₯ ) = βˆ’ ( π‘₯ βˆ’ 6 ) + 1 2
  • D 𝑓 ( π‘₯ ) = βˆ’ ( π‘₯ βˆ’ 3 ) + 1 4 2
  • E 𝑓 ( π‘₯ ) = ( π‘₯ βˆ’ 3 ) + 1 4 2

Q7:

In completing the square for quadratic function 𝑓 ( π‘₯ ) = π‘₯ + 1 4 π‘₯ + 4 6 2 , you arrive at the expression ( π‘₯ βˆ’ 𝑏 ) + 𝑐 2 . What is the value of 𝑐 ?

Q8:

Write the equation π‘₯ + 6 π‘₯ βˆ’ 3 = 0 2 in completed square form.

  • A ( π‘₯ + 3 ) βˆ’ 9 = 0 2
  • B ( π‘₯ βˆ’ 3 ) βˆ’ 1 2 = 0 2
  • C ( π‘₯ βˆ’ 3 ) βˆ’ 9 = 0 2
  • D ( π‘₯ + 3 ) βˆ’ 1 2 = 0 2
  • E ( π‘₯ + 6 ) βˆ’ 3 9 = 0 2

Q9:

Write the equation 1 + π‘₯ = π‘₯ 2 in the form ( π‘₯ βˆ’ 𝑝 ) = π‘ž 2 .

  • A ο€Ό π‘₯ + 1 2  = 5 4 2
  • B ο€Ό π‘₯ βˆ’ 1 4  = 5 4 2
  • C ο€Ό π‘₯ βˆ’ 1 2  = 1 2
  • D ο€Ό π‘₯ βˆ’ 1 2  = 5 4 2
  • E ο€Ό π‘₯ βˆ’ 1 4  = 1 2

Q10:

Write the equation π‘₯ + π‘₯ + 1 = 0 2 in the form ( π‘₯ βˆ’ 𝑝 ) = π‘ž 2 .

  • A ο€Ό π‘₯ + 1 2  = βˆ’ 1 2
  • B ο€Ό π‘₯ + 1 4  = βˆ’ 3 4 2
  • C ο€Ό π‘₯ βˆ’ 1 2  = βˆ’ 3 4 2
  • D ο€Ό π‘₯ + 1 2  = βˆ’ 3 4 2
  • E ο€Ό π‘₯ + 1 4  = βˆ’ 1 2

Q11:

Write the equation 3 π‘₯ βˆ’ 1 = 0 2 in the form ( π‘₯ βˆ’ 𝑝 ) = π‘ž 2 .

  • A ο€Ό π‘₯ βˆ’ 1 3  = 1 9 2
  • B ο€Ό π‘₯ βˆ’ 1 3  = 0 2
  • C π‘₯ = 1 9 2
  • D π‘₯ = 1 3 2

Q12:

Write the equation π‘₯ = 3 0 βˆ’ 1 3 π‘₯ 2 in the form ( π‘₯ βˆ’ 𝑝 ) = π‘ž 2 .

  • A ο€Ό π‘₯ + 1 3 2  = 3 0 2
  • B ο€Ό π‘₯ βˆ’ 1 3 2  = 2 8 9 4 2
  • C ο€Ό π‘₯ + 1 6 9 4  = 2 8 9 4 2
  • D ο€Ό π‘₯ + 1 3 2  = 2 8 9 4 2
  • E ο€Ό π‘₯ + 1 6 9 4  = 3 0 2

Q13:

Write the equation π‘₯ βˆ’ π‘₯ = 3 4 2 in the form ( π‘₯ βˆ’ 𝑝 ) = π‘ž 2 .

  • A ο€Ό π‘₯ + 1 2  = 1 2
  • B ο€Ό π‘₯ βˆ’ 1 4  = 1 2
  • C ο€Ό π‘₯ βˆ’ 1 2  = 3 4 2
  • D ο€Ό π‘₯ βˆ’ 1 2  = 1 2
  • E ο€Ό π‘₯ βˆ’ 1 4  = 3 4 2

Q14:

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

  • A ο€» π‘₯ βˆ’ √ 3  = βˆ’ 2 2
  • B ( π‘₯ βˆ’ 3 ) = 2 2
  • C ο€» π‘₯ βˆ’ √ 3  = βˆ’ 1 2
  • D ο€» π‘₯ βˆ’ √ 3  = 2 2
  • E ( π‘₯ + 3 ) = 2 2

Q15:

Write the equation 3 π‘₯ + 𝑏 π‘₯ βˆ’ 1 = 0 2 in the form ( π‘₯ βˆ’ 𝑝 ) = π‘ž 2 .

  • A ο€½ π‘₯ + 𝑏 6  = 1 2
  • B ο€½ π‘₯ βˆ’ 𝑏 6  = 𝑏 + 1 2 3 6 2 2
  • C ο€Ύ π‘₯ + 𝑏 3 6  = 𝑏 + 1 2 3 6 2 2 2
  • D ο€½ π‘₯ + 𝑏 6  = 𝑏 + 1 2 3 6 2 2
  • E ο€Ύ π‘₯ βˆ’ 𝑏 3 6  = 𝑏 + 1 2 3 6 2 2 2

Q16:

Write the equation 3 π‘₯ + 𝑏 π‘₯ + 𝑐 = 0 2 in the form ( π‘₯ βˆ’ 𝑝 ) = π‘ž 2 .

  • A ο€½ π‘₯ βˆ’ 𝑏 6  = 𝑏 βˆ’ 1 2 𝑐 3 6 2 2
  • B ο€½ π‘₯ + 𝑏 3  = 𝑏 βˆ’ 3 𝑐 9 2 2
  • C ο€½ π‘₯ + 𝑏 6  = βˆ’ 𝑐 3 2
  • D ο€½ π‘₯ + 𝑏 6  = 𝑏 βˆ’ 1 2 𝑐 3 6 2 2
  • E ο€½ π‘₯ βˆ’ 𝑏 3  = 𝑏 βˆ’ 3 𝑐 9 2 2

Q17:

Write the equation π‘₯ + 𝑏 π‘₯ + 𝑐 = 0 2 in the form ( π‘₯ βˆ’ 𝑝 ) = π‘ž 2 .

  • A ο€½ π‘₯ + 𝑏 2  = βˆ’ 𝑐 2
  • B ο€½ π‘₯ βˆ’ 𝑏 2  = 𝑏 βˆ’ 4 𝑐 4 2 2
  • C ο€Ύ π‘₯ + 𝑏 4  = 𝑏 βˆ’ 4 𝑐 4 2 2 2
  • D ο€½ π‘₯ + 𝑏 2  = 𝑏 βˆ’ 4 𝑐 4 2 2
  • E ο€½ π‘₯ + 𝑏 2  = 𝑏 + 4 𝑐 4 2 2

Q18:

Write the equation π‘Ž π‘₯ + 𝑏 π‘₯ + 𝑐 = 0 2 , where π‘Ž β‰  0 , in the form ( π‘₯ βˆ’ 𝑝 ) = π‘ž 2 .

  • A ο€½ π‘₯ + 𝑏 2 π‘Ž  = βˆ’ 𝑐 π‘Ž 2
  • B ο€½ π‘₯ βˆ’ 𝑏 2 π‘Ž  = 𝑏 βˆ’ 4 π‘Ž 𝑐 4 π‘Ž 2 2 2
  • C ο€½ π‘₯ + 𝑏 2 π‘Ž  = 𝑏 βˆ’ π‘Ž 𝑐 π‘Ž 2 2 2
  • D ο€½ π‘₯ + 𝑏 2 π‘Ž  = 𝑏 βˆ’ 4 π‘Ž 𝑐 4 π‘Ž 2 2 2
  • E ο€½ π‘₯ βˆ’ 𝑏 2 π‘Ž  = βˆ’ 𝑐 π‘Ž 2

Q19:

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

  • A ( π‘₯ + 7 ) = βˆ’ 3 2
  • B ( π‘₯ βˆ’ 7 ) = 4 6 2
  • C ( π‘₯ + 4 9 ) = 4 6 2
  • D ( π‘₯ + 7 ) = 4 6 2
  • E ( π‘₯ + 4 9 ) = βˆ’ 3 2

Q20:

Given that π‘₯ βˆ’ π‘₯ βˆ’ 𝑐 = 0 2 can be written in the form ( π‘₯ βˆ’ 𝑝 ) = 3 2 , find the value of 𝑐 .

  • A βˆ’ 1 3 4
  • B 1 3 4
  • C3
  • D 1 1 4
  • E βˆ’ 1 1 4

Q21:

Find the values of for which the equation is satisfied by only one value of .

  • A
  • B
  • C
  • D
  • E

Q22:

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

Q23:

Which of the following equations can be expanded and rearranged to π‘₯ + 1 = 8 π‘₯ 2 ?

  • A ( π‘₯ βˆ’ 4 ) = βˆ’ 1 5 2
  • B ( π‘₯ + 4 ) = 1 5 2
  • C ( π‘₯ + 4 ) = βˆ’ 1 5 2
  • D ( π‘₯ βˆ’ 4 ) = 1 5 2
  • E ( π‘₯ βˆ’ 8 ) = 1 5 2

Q24:

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

  • A when π‘Ž > 0
  • B when π‘Ž < 0 or π‘Ž > 0
  • C when π‘Ž < 0
  • D when 0 < π‘Ž < 1
  • Ewhen π‘Ž < 1

Q25:

Completely factor .

  • A
  • B
  • C
  • D
  • E