Worksheet: Factoring Quadratics

In this worksheet, we will practice factoring (if possible) quadratic equations with one or two variables as perfect squares, differences of squares, and trinomials.

Q1:

Factorise fully 6 4 ( 𝑥 + 1 ) 9 ( 𝑥 1 ) 2 2 .

  • A ( 1 1 𝑥 + 1 1 ) ( 5 𝑥 + 5 )
  • B ( 1 1 𝑥 + 8 ) ( 5 𝑥 + 3 )
  • C ( 8 𝑥 + 1 1 ) ( 3 𝑥 + 5 )
  • D ( 5 𝑥 + 1 1 ) ( 1 1 𝑥 + 5 )
  • E 2 4 ( 𝑥 + 1 ) ( 𝑥 1 )

Q2:

Factor 𝑥 8 𝑥 + 1 2 2 .

  • A ( 𝑥 + 6 ) ( 𝑥 + 2 )
  • B ( 𝑥 3 ) ( 𝑥 4 )
  • C ( 𝑥 + 3 ) ( 𝑥 + 4 )
  • D ( 𝑥 6 ) ( 𝑥 2 )
  • E ( 𝑥 + 6 ) ( 𝑥 2 )

Q3:

Factor 6 𝑥 𝑥 1 2 2 .

  • A ( 2 𝑥 + 3 ) ( 3 𝑥 + 4 )
  • B ( 2 𝑥 + 3 ) ( 3 𝑥 4 )
  • C ( 2 𝑥 3 ) ( 3 𝑥 4 )
  • D ( 2 𝑥 3 ) ( 3 𝑥 + 4 )
  • E ( 𝑥 4 ) ( 𝑥 + 3 )

Q4:

Factorise fully 4 9 𝑥 5 6 𝑥 𝑦 + 1 6 𝑦 2 2 4 .

  • A ( 7 4 𝑥 𝑦 ) 2
  • B ( 7 𝑥 4 𝑦 ) 2
  • C ( 7 4 𝑥 𝑦 ) 2 2
  • D ( 7 𝑥 4 𝑦 ) 2 2
  • E ( 7 𝑥 + 4 𝑦 ) 2 2

Q5:

Factorise fully ( 4 𝑥 + 7 ) ( 5 𝑥 2 ) 2 2 .

  • A ( 9 𝑥 + 5 ) 2
  • B ( 9 𝑥 + 5 ) ( 1 1 𝑥 + 9 )
  • C 9 ( 𝑥 + 1 ) ( 1 1 𝑥 + 1 2 )
  • D ( 9 𝑥 + 5 ) ( 9 𝑥 )
  • E 9 ( 𝑥 1 ) ( 1 1 𝑥 + 1 2 )

Q6:

Complete the following: 𝑥 3 7 𝑥 + 5 6 = ( 3 𝑥 8 ) ( ) 2 .

  • A2, 2 𝑥 7
  • B6, 2 𝑥 + 7
  • C3, 2 𝑥 7
  • D6, 2 𝑥 7
  • E6, 𝑥 7

Q7:

Expand and simplify 6 𝑥 + 𝑦 ( 1 0 𝑦 1 9 𝑥 ) 2 , then factorise the result.

  • A ( 2 𝑥 + 5 𝑦 ) ( 3 𝑥 2 𝑦 )
  • B ( 2 𝑥 + 5 𝑦 ) ( 3 𝑥 + 2 𝑦 )
  • C ( 6 𝑥 5 𝑦 ) ( 𝑥 2 𝑦 )
  • D ( 2 𝑥 5 𝑦 ) ( 3 𝑥 2 𝑦 )

Q8:

Expand and simplify 𝑎 ( 𝑎 + 9 ) + 1 8 , then factorise the result.

  • A ( 𝑎 + 9 ) ( 𝑎 + 2 )
  • B 𝑎 + 6 𝑎 + 3 2 2
  • C ( 𝑎 + 1 8 ) ( 𝑎 + 1 )
  • D ( 𝑎 + 6 ) ( 𝑎 + 3 )

Q9:

Factorise fully 𝑥 1 9 𝑥 + 8 4 6 3 .

  • A 𝑥 + 3 𝑥 + 2 8 3 3
  • B 𝑥 1 2 𝑥 7 2 2
  • C 𝑥 + 4 𝑥 + 2 1 3 3
  • D 𝑥 1 2 𝑥 7 3 3

Q10:

Factorise fully 𝑥 + 𝑥 + 1 2 2 .

  • A ( 𝑥 + 4 ) ( 𝑥 + 3 )
  • B ( 𝑥 + 4 ) ( 𝑥 3 )
  • C ( 𝑥 4 ) ( 𝑥 3 )
  • D ( 𝑥 4 ) ( 𝑥 + 3 )
  • E ( 𝑥 + 6 ) ( 𝑥 2 )

Q11:

Factorise fully 𝑎 + 6 𝑎 1 0 ( 𝑎 + 6 ) 2 .

  • A ( 𝑎 + 1 0 ) ( 𝑎 6 )
  • B 𝑎 1 0 𝑎 + 6 2 2
  • C ( 𝑎 + 3 ) ( 𝑎 2 0 )
  • D ( 𝑎 1 0 ) ( 𝑎 + 6 )

Q12:

Complete the following: 9 𝑥 + 3 2 𝑥 𝑦 + = ( 𝑥 + 4 𝑦 ) ( ) 2 .

  • A 1 6 𝑦 2 , 9 𝑥 4
  • B 1 6 , 9 𝑥 4 𝑦
  • C 1 6 𝑦 2 , 4 𝑥 + 9 𝑦
  • D 1 6 𝑦 2 , 9 𝑥 4 𝑦
  • E 1 6 , 4 𝑥 + 9 𝑦

Q13:

Factorise fully 6 0 4 ( 𝑥 𝑦 ) ( 𝑥 𝑦 ) 2 .

  • A ( 𝑥 𝑦 1 0 ) ( 𝑥 𝑦 + 6 )
  • B ( 𝑥 + 𝑦 + 1 0 ) ( 𝑥 + 𝑦 6 )
  • C ( 𝑥 𝑦 + 3 ) ( 𝑥 𝑦 2 0 )
  • D ( 𝑥 𝑦 + 1 0 ) ( 𝑥 𝑦 6 )
  • E ( 𝑥 + 𝑦 + 4 ) ( 𝑥 + 𝑦 1 5 )

Q14:

Complete the factorisation: 2 0 𝑥 2 1 𝑥 + 4 = ( 4 𝑥 ) ( ) 2 .

  • A ( 4 𝑥 1 ) ( 5 𝑥 5 )
  • B ( 4 𝑥 2 ) ( 5 𝑥 4 )
  • C ( 4 𝑥 1 ) ( 𝑥 4 )
  • D ( 4 𝑥 1 ) ( 5 𝑥 4 )
  • E ( 4 𝑥 1 ) ( 2 𝑥 5 )

Q15:

Complete the following: 𝑧 + 1 6 = ( 𝑧 + ) ( + 2 ) 2 .

  • A 6 𝑧 , 8, 𝑧
  • B 6 𝑧 , 8 , 𝑧
  • C 6 𝑧 , 8, 8 𝑧
  • D 6 𝑧 , 8 , 𝑧
  • E 6 𝑧 , 14, 8 𝑧

Q16:

Expand and simplify ( 𝑎 5 𝑏 ) ( 𝑎 + 5 𝑏 ) + 2 4 𝑎 𝑏 , then factorise the result completely.

  • A ( 𝑎 + 1 ) ( 𝑎 2 5 )
  • B ( 𝑎 1 ) ( 𝑎 + 2 5 )
  • C ( 𝑎 + 5 𝑏 ) ( 𝑎 5 𝑏 )
  • D ( 𝑎 𝑏 ) ( 𝑎 + 2 5 𝑏 )

Q17:

If 𝑎 1 0 𝑎 𝑏 + 2 1 𝑏 = 3 0 2 2 and 𝑎 3 𝑏 = 3 , what is the value of 𝑎 7 𝑏 ?

Q18:

The expression 𝑥 + 𝑎 𝑥 1 8 2 can be factorised. Given that 𝑎 is a negative integer, find the set of values of 𝑎 .

  • A { 1 , 1 8 , 2 , 9 , 3 , 6 }
  • B { 1 7 , 7 , 3 , 3 , 7 , 1 7 }
  • C { 3 , 6 }
  • D { 1 7 , 7 , 3 }
  • E { 3 , 6 }

Q19:

The area of a rectangle is 5 𝑥 + 1 2 𝑥 + 7 2 cm2. Find its dimensions in terms of 𝑥 and its perimeter when 𝑥 = 4 .

  • A ( 5 𝑥 1 ) cm, ( 𝑥 + 7 ) cm, 60 cm
  • B ( 5 𝑥 7 ) cm, ( 𝑥 + 1 ) cm, 36 cm
  • C ( 5 𝑥 1 ) cm, ( 𝑥 7 ) cm, 32 cm
  • D ( 5 𝑥 + 7 ) cm, ( 𝑥 + 1 ) cm, 64 cm

Q20:

Factorise fully 1 2 𝑥 𝑦 2 6 𝑥 𝑦 + 1 2 𝑥 𝑦 2 2 2 3 2 4 .

  • A 𝑥 𝑦 ( 4 𝑥 + 3 ) ( 3 𝑥 2 ) 2 2
  • B 2 𝑥 𝑦 ( 2 𝑦 + 3 ) ( 3 𝑦 + 2 ) 2 2
  • C 2 𝑥 𝑦 ( 𝑦 3 ) ( 6 𝑦 2 ) 2 2
  • D 2 𝑦 𝑥 ( 2 𝑦 3 ) ( 3 𝑦 2 ) 2 2
  • E 𝑥 𝑦 ( 1 2 𝑥 3 ) ( 𝑥 2 ) 2 2

Q21:

Complete the following: 8 𝑥 = ( 4 𝑥 + 7 𝑦 ) ( + 5 𝑦 ) 2 .

  • A + 3 4 𝑥 𝑦 + 3 5 2 ,
  • B + 3 5 𝑥 𝑦 + 3 4 𝑦 2 2 ,
  • C 3 4 𝑥 𝑦 + 3 5 𝑦 2 𝑥 , 2
  • D + 3 4 𝑥 𝑦 + 3 5 𝑦 2 𝑥 2 ,
  • E + 3 4 𝑥 𝑦 + 3 5 𝑦 2 2 ,

Q22:

Factorise fully 𝑥 + 1 0 𝑥 𝑦 + 1 6 𝑦 4 2 2 .

  • A 𝑥 8 𝑦 𝑥 + 2 𝑦 2 2
  • B 𝑥 8 𝑦 𝑥 2 𝑦 2 2
  • C 𝑥 + 4 𝑦 2 2
  • D 𝑥 + 8 𝑦 𝑥 + 2 𝑦 2 2

Q23:

Expand and simplify 1 5 𝑥 2 𝑦 ( 4 𝑦 7 𝑥 ) 2 , then factorise the result completely.

  • A ( 3 𝑥 4 𝑦 ) ( 5 𝑥 2 𝑦 )
  • B ( 3 𝑥 4 𝑦 ) ( 5 𝑥 + 2 𝑦 )
  • C ( 1 5 𝑥 + 4 𝑦 ) ( 𝑥 2 𝑦 )
  • D ( 3 𝑥 + 4 𝑦 ) ( 5 𝑥 2 𝑦 )

Q24:

Factorise fully 𝑦 5 𝑦 1 4 4 2 .

  • A ( 𝑦 2 ) ( 𝑦 7 ) 2 2
  • B ( 𝑦 2 ) ( 𝑦 + 7 ) 2 2
  • C ( 𝑦 + 1 ) ( 𝑦 1 4 ) 2 2
  • D ( 𝑦 + 2 ) ( 𝑦 7 ) 2 2

Q25:

Given that the expression 𝑥 + 𝑎 𝑥 2 8 2 can be factorised, what is the set of values of 𝑎 .

  • A { 1 , 2 8 , 2 , 1 4 , 4 , 7 }
  • B { 2 7 , 1 2 , 3 }
  • C { 7 , 4 }
  • D { 2 7 , 1 2 , 3 , 3 , 1 2 , 2 7 }
  • E { 7 , 4 }

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