Worksheet: Factoring Perfect Trinomials

In this worksheet, we will practice identifying and factoring quadratics that are perfect trinomials and completing the factorization.

Q1:

For which values of 𝑘 is 1 6 𝑥 + 𝑘 𝑥 + 8 1 a perfect square?

  • A 2 6 , 2 6
  • B 1 3 , 1 3
  • C 3 6 , 3 6
  • D 7 2 , 7 2

Q2:

Complete the expression 1 6 𝑥 + 𝑦 to make a perfect square.

  • A + 8 𝑥 𝑦 8 𝑥 𝑦 o r
  • B + 1 0 𝑥 𝑦 1 0 𝑥 𝑦 o r
  • C + 8 𝑥 𝑦 8 𝑥 𝑦 o r
  • D + 4 𝑥 𝑦 4 𝑥 𝑦 o r

Q3:

Which of the following is a perfect square?

  • A 𝑥 8 1
  • B 𝑥 1 8 𝑥 8 1
  • C 𝑥 1 8 𝑥 + 8 1
  • D 𝑥 9 𝑥 + 8 1
  • E 𝑥 + 8 1

Q4:

Complete the quadratic expression 9 𝑥 + 1 4 4 to make a perfect square.

  • A + 7 2 𝑥 7 2 𝑥 o r
  • B + 3 6 𝑥 3 6 𝑥 o r
  • C + 7 2 𝑥 7 2 𝑥 o r
  • D + 7 2 𝑥
  • E + 3 0 𝑥

Q5:

If 9 𝑦 + 3 0 𝑦 + 𝑏 is a perfect square, what is the value of 𝑏 ?

Q6:

Complete the expression 6 0 𝑥 + 2 5 to make a perfect square.

  • A 3 6 𝑥
  • B 6 𝑥
  • C 1 2 𝑥
  • D 1 2 𝑥
  • E 3 6 𝑥

Q7:

Complete the expression 4 2 5 𝑎 + 1 9 𝑏 to make a perfect square.

  • A ± 2 1 5 𝑎 𝑏
  • B + 2 5 𝑎 𝑏
  • C + 4 5 𝑎 𝑏
  • D ± 4 1 5 𝑎 𝑏
  • E ± 4 1 5 𝑎 𝑏

Q8:

If 𝑘 𝑦 2 4 𝑦 + 9 is a perfect square, what is the value of 𝑘 ?

Q9:

Factorise fully 9 𝑥 + 3 6 𝑥 𝑦 + 3 6 𝑦 .

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

Q10:

Factorise fully 𝑥 1 0 𝑥 𝑦 + 2 5 𝑦 .

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

Q11:

By factorising or otherwise, evaluate ( 1 0 . 1 ) 4 . 2 × 1 0 . 1 + ( 2 . 1 ) .

Q12:

Factorise fully 8 1 𝑚 + 1 8 𝑚 + 1 .

  • A ( 9 𝑚 + 1 ) ( 9 𝑚 1 )
  • B ( 9 𝑚 + 1 ) ( 𝑚 + 9 )
  • C ( 9 𝑚 + 1 )
  • D ( 𝑚 9 )
  • E ( 𝑚 + 9 )

Q13:

Factorise fully 4 ( 𝑥 7 ) 1 6 ( 𝑥 7 ) ( 𝑦 6 ) + 1 6 ( 𝑦 6 ) .

  • A 2 𝑥 4 𝑦 + 1 0
  • B ( 2 𝑥 4 𝑦 3 8 )
  • C ( 𝑥 + 𝑦 2 )
  • D ( 2 𝑥 4 𝑦 )
  • E ( 2 𝑥 4 𝑦 + 1 0 )

Q14:

Complete the expression 8 1 𝑥 + 9 0 𝑥 𝑦 + to make a perfect square.

  • A 2 0 𝑦
  • B 2 5 𝑦
  • C 9 𝑦
  • D 2 5 𝑦
  • E 2 5 𝑦

Q15:

Factorise fully 4 9 8 1 𝑥 + 8 9 𝑥 + 1 6 4 9 .

  • A 7 9 𝑥 + 4 7
  • B 7 9 𝑥 + 2 7
  • C 7 9 𝑥 2 7
  • D 7 9 𝑥 + 4 7
  • E 7 9 𝑥 4 7

Q16:

Expand and simplify 5 𝑥 ( 5 𝑥 + 1 8 𝑦 ) + 8 1 𝑦 , then factorise the result.

  • A ( 5 𝑥 + 9 𝑦 )
  • B ( 5 𝑥 9 𝑦 )
  • C ( 5 𝑥 + 8 𝑦 )
  • D ( 5 𝑥 + 9 )
  • E ( 5 𝑥 + 8 )

Q17:

By factorising or otherwise, evaluate ( 9 7 ) + 6 × 9 7 + 9 .

Q18:

Factorise fully 0 . 1 6 𝑏 + 7 . 2 𝑏 + 8 1 .

  • A ( 0 . 2 𝑏 + 9 )
  • B ( 0 . 4 𝑏 + 9 )
  • C ( 0 . 4 + 9 𝑏 )
  • D ( 0 . 2 + 9 𝑏 )
  • E ( 0 . 4 ( 𝑏 + 9 ) )

Q19:

Factorise fully 4 8 1 𝑥 4 3 𝑥 + 9 .

  • A 2 9 𝑥 3
  • B ( 9 𝑥 3 )
  • C 2 9 𝑥 + 3 2 9 𝑥 3
  • D 3 𝑥 + 2 9 3 𝑥 2 9
  • E 3 𝑥 2 9

Q20:

Given that 𝑥 + 2 𝑥 𝑦 + 𝑦 = 8 1 , what are the possible values of 𝑥 + 𝑦 ?

  • A 9 , 9
  • B 1 8 , 1 8
  • C 4 0 , 4 0
  • D 8 1 , 8 1

Q21:

Factorise fully 𝑎 + 6 𝑎 + 9 𝑎 .

  • A ( 𝑎 + 3 )
  • B 𝑎 ( 𝑎 + 3 )
  • C 𝑎 ( 3 𝑎 + 1 )
  • D 𝑎 ( 3 𝑎 + 1 )
  • E 𝑎 ( 3 𝑎 + 1 ) ( 3 𝑎 1 )

Q22:

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

  • A 1 6 𝑦 ( 2 𝑥 + 1 )
  • B 1 6 𝑦 ( 8 4 𝑥 )
  • C 1 6 𝑦 ( 2 𝑥 1 )
  • D 1 6 ( 8 𝑥 + 4 ) ( 8 𝑥 4 )
  • E 1 6 𝑥 ( 8 4 𝑦 )

Q23:

Write all possible expressions for 𝑘 such that 𝑥 + 𝑘 + 4 𝑦 is a perfect square.

  • A 4 , 4
  • B 6 𝑥 𝑦 , 6 𝑥 𝑦
  • C 4 𝑥 𝑦 , 4 𝑥 𝑦
  • D 2 𝑥 𝑦 , 2 𝑥 𝑦

Q24:

Factorise fully ( 𝑎 + 4 𝑏 ) + 1 4 𝑐 ( 𝑎 + 4 𝑏 ) + 4 9 𝑐 .

  • A 𝑎 + 4 𝑏 + 4 9 𝑐
  • B 𝑎 + 4 𝑏 7 𝑐
  • C 𝑎 + 4 𝑏 + 1 4 𝑐
  • D 𝑎 + 4 𝑏 + 7 𝑐
  • E ( 𝑎 + 4 𝑏 + 7 𝑐 )

Q25:

Expand and simplify ( 3 𝑥 + 𝑦 ) 1 2 𝑥 𝑦 , and then factorise the result completely.

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

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