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?

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

Q2:

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

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

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

Q4:

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

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

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?

  • 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

Q7:

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

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

Q8:

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

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

Q9:

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

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

Q10:

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

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

Q11:

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

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

Q12:

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

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

Q13:

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

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

Q14:

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

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

Q15:

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

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

Q16:

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

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

Q17:

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

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

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

Q19:

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

  • A βˆ’ 1 3 4
  • B 1 1 4
  • C βˆ’ 1 1 4
  • D 1 3 4
  • 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‘₯.

  • A π‘Ž = βˆ’ 2 + √ 5 , π‘Ž = 0 , π‘Ž = βˆ’ 2 + √ 5
  • B π‘Ž = 1 , π‘Ž = 0
  • C π‘Ž = 1 , π‘Ž = βˆ’ 1
  • D π‘Ž = 1 , π‘Ž = 0 , π‘Ž = βˆ’ 1
  • E π‘Ž = 0 , π‘Ž = 1 βˆ’ √ 5 2 , π‘Ž = 1 + √ 5 2

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

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:

Factorise fully 8𝑦𝑛+162𝑧𝑛οŠͺοŠͺ.

  • A 2 𝑛 ο€Ή 2 𝑦 + 9 𝑧  βˆ’ 6 𝑦 𝑧     οŠͺ οŠͺ
  • B 2 𝑛 ο€Ή 2 𝑦 βˆ’ 6 𝑦 𝑧 + 9 𝑧  ο€Ή 2 𝑦 + 6 𝑦 𝑧 + 9 𝑧      
  • C 2 𝑛 ο€Ή 2 𝑦 + 9 𝑧  ο€Ή 2 𝑦 βˆ’ 9 𝑧      
  • D 2 𝑛 ο€Ή 2 𝑦 + 9 𝑧     
  • E 2 𝑛 ο€Ή 2 𝑦 βˆ’ 1 8 𝑦 𝑧 + 9 𝑧  ο€Ή 2 𝑦 + 1 8 𝑦 𝑧 + 9 𝑧      

Q25:

Factorise fully 4π‘₯+9+8π‘₯οŠͺ.

  • A ο€Ή 2 π‘₯ βˆ’ 2 π‘₯ βˆ’ 3  ο€Ή 2 π‘₯ + 2 π‘₯ βˆ’ 3   
  • B ο€Ή 2 π‘₯ βˆ’ 2 π‘₯ βˆ’ 3  ο€Ή 2 π‘₯ + 2 π‘₯ βˆ’ 3     
  • C ο€Ή 2 π‘₯ βˆ’ 4 π‘₯ + 3  ο€Ή 2 π‘₯ + 4 π‘₯ + 3   
  • D ο€Ή 2 π‘₯ βˆ’ 2 π‘₯ + 3  ο€Ή 2 π‘₯ + 2 π‘₯ + 3   
  • E ο€Ή 2 π‘₯ βˆ’ 4 π‘₯ βˆ’ 3   

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