Worksheet: Substitution to Manipulate Expressions into Quadratics

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In this worksheet, we will practice solving complicated equations by making substitutions that reduce the equation to a quadratic.

Q1:

Factorise fully ( 𝑧 + 3 ) − 1 9 ( 𝑧 + 3 ) + 9 0 .

  • A ( 𝑧 + 3 ) ( 𝑧 + 3 0 )
  • B ( 𝑧 − 9 ) ( 𝑧 − 1 0 )
  • C ( 𝑧 + 8 ) ( 𝑧 + 2 1 )
  • D ( 𝑧 − 6 ) ( 𝑧 − 7 )

Q2:

Find the solution set of 5 + 1 2 5 5 = 3 0 .

  • A { 4 , 4 4 }
  • B { 5 , 2 5 }
  • C { 1 0 , 1 0 0 }
  • D { 1 , 2 }

Q3:

Find the solution set of 𝑥 − 3 3 𝑥 + 3 2 = 0 .

  • A { 1 , 2 }
  • B { 4 , 1 6 }
  • C { 2 }
  • D { 1 , 4 }
  • E { 1 , 3 2 }

Q4:

Factorise fully 2 4 𝑥 − 3 4 𝑥 + 1 2 𝑥 .

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

Q5:

Find the solution set of 2 − 2 0 × 2 + 2 5 6 = 0 .

  • A { 3 , − 3 }
  • B { 4 , 1 6 }
  • C { 6 , − 6 }
  • D { 2 , 4 }
  • E { 2 , − 2 }

Q6:

Find the solution set of 𝑥 − 7 3 0 𝑥 + 7 2 9 = 0 .

  • A { 5 , 1 5 , − 5 , − 1 5 }
  • B { 1 , 3 , − 1 , − 3 }
  • C { − 1 , − 3 }
  • D { 1 , 2 4 3 , − 1 , − 2 4 3 }
  • E { − 5 , − 1 5 }

Q7:

Consider the equation 𝑥 + 2 = 2 𝑥 . Adding 𝑥 + 𝑥 to both sides produces

Is the solution set of the equation formed the same as that of the original equation?

  • Ayes
  • Bno

Q8:

Find the solution set of 3 + 7 2 9 3 = 9 0 .

  • A { 8 , 8 4 }
  • B { 9 , 8 1 }
  • C { 3 6 , 3 6 0 }
  • D { 2 , 4 }

Q9:

Factorise fully ( 𝑥 − 7 ) + 6 ( 𝑥 − 7 ) − 7 2 .

  • A ( 𝑥 − 1 9 ) ( 𝑥 − 1 )
  • B ( 𝑥 + 1 2 ) ( 𝑥 − 6 )
  • C ( 𝑥 + 3 ) ( 𝑥 − 2 4 )
  • D ( 𝑥 + 5 ) ( 𝑥 − 1 3 )
  • E ( 𝑥 − 3 ) ( 𝑥 − 2 5 )

Q10:

Factorise fully ( 𝑦 − 4 ) − 1 8 ( 𝑦 − 4 ) + 8 0 .

  • A ( 𝑦 + 4 ) ( 𝑦 + 2 0 )
  • B ( 𝑦 − 1 0 ) ( 𝑦 − 8 )
  • C ( 𝑦 + 1 ) ( 𝑦 + 1 2 )
  • D ( 𝑦 − 1 4 ) ( 𝑦 − 1 2 )

Q11:

Find the solution set of 𝑥 − 9 𝑥 + 1 4 = 0 .

  • A { 2 1 , 6 }
  • B { 7 , 2 }
  • C { − 7 , − 2 }
  • D { 3 4 3 , 8 }
  • E { − 2 1 , − 6 }

Q12:

Find the solution set of 2 − 1 0 × 2 + 3 2 = 0 .

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

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