Worksheet: Factoring Nonmonic Quadratics

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In this worksheet, we will practice factoring quadratics where the coefficient of the leading term is greater than one.

Q1:

Factorise fully 4 𝑥 − 3 2 𝑥 + 2 8 .

  • A 4 ( 𝑥 + 1 ) ( 𝑥 + 7 )
  • B 4 ( 𝑥 − 1 ) ( 𝑥 + 7 )
  • C ( 4 𝑥 + 1 ) ( 𝑥 − 7 )
  • D 4 ( 𝑥 − 1 ) ( 𝑥 − 7 )

Q2:

Factorise fully 6 𝑥 − 1 9 𝑥 + 1 0 .

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

Q3:

Solve the equation 4 𝑡 − 3 2 𝑡 + 6 4 = 0 by factoring.

  • A 𝑡 = − 4
  • B 𝑡 = 4
  • C 𝑡 = 8 or 𝑡 = 2
  • D 𝑡 = − 8 or 𝑡 = − 2

Q4:

Find the solution set of the equation 1 8 𝑥 + 1 8 𝑥 − 3 6 = 0 in ℝ .

  • A { − 2 , 1 }
  • B { 4 , − 9 }
  • C { − 4 , − 9 }
  • D { 2 , − 1 }

Q5:

Find the solution set of 6 𝑥 − 1 3 𝑥 + 6 = 0 in ℝ .

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

Q6:

Given that − 1 0 is a root of the equation 2 𝑥 + 1 3 𝑥 − 7 0 = 0 , what is the other root?

  • A − 2
  • B 2 7
  • C 7 2
  • D2
  • E − 7 2

Q7:

Find the solution set of 2 𝑥 + 5 𝑥 − 7 = 0 in ℝ .

  • A 7 2 , − 1
  • B − 7 2 , 1
  • C − 2 7 , − 1
  • D { − 7 , 1 }
  • E { 7 , − 1 }

Q8:

Solve the equation 2 ( 𝑥 + 1 ) + 5 ( 𝑥 + 1 ) = 0 .

  • A 𝑥 = − 1 , 𝑥 = − 7 2
  • B 𝑥 = 5 2
  • C 𝑥 = − 1 , 𝑥 = − 5 2
  • D 𝑥 = − 5 2
  • E 𝑥 = 1

Q9:

Find the solution set of ( 2 𝑦 + 4 ) + ( 𝑦 + 2 ) = 5 in ℝ .

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

Q10:

Find the solution set of 𝑥 ( 𝑥 + 5 ) 4 − 𝑥 ( 𝑥 + 1 ) 8 − 3 ( 𝑥 + 4 ) 2 + 1 = 0 in ℝ .

  • A { 1 , − 4 0 }
  • B { − 5 , 8 }
  • C { 5 , − 8 }
  • D { 2 , − 2 0 }
  • E { − 1 , 4 0 }

Q11:

Find the solution set of − 7 ( 𝑥 + 7 ) + 9 ( 𝑥 + 7 ) = 0 in ℝ .

  • A − 7 , 5 8 7
  • B − 4 0 7
  • C − 7 , − 4 0 7
  • D 7 , − 4 0 7
  • E 0 , − 4 0 7

Q12:

The roots of the equation 𝑥 − 1 0 𝑥 + 1 6 = 0 are 𝐿 and 𝑀 , where 𝐿 > 𝑀 . Find, in its simplest form, the quadratic equation whose roots are 𝐿 − 7 and 2 𝑀 − 6 .

  • A 𝑥 − 9 𝑥 − 1 0 = 0
  • B 𝑥 + 𝑥 − 2 = 0
  • C 𝑥 + 3 𝑥 + 2 = 0
  • D 𝑥 + 𝑥 − 6 = 0
  • E 𝑥 − 3 𝑥 + 2 = 0

Q13:

Find the solution set of the equation 3 𝑥 − 9 𝑥 + 6 = 0 , giving values to the nearest tenth.

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

Q14:

Find the solution set of 𝑥 + 9 3 𝑥 + 3 8 = 1 2 in ℝ .

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

Q15:

Find the solution set of ( 3 𝑥 + 6 ) = ( 5 𝑥 − 1 1 ) in ℝ .

  • A 1 7 2 , − 1 7 2
  • B − 1 1 5 , 2
  • C 1 7 2
  • D 1 1 5 , − 2
  • E 1 7 2 , 5 8

Q16:

Find the solution set of 1 4 4 𝑥 = 3 6 in ℝ .

  • A 1 2
  • B − 1 2
  • C − 1 2 , 1 2
  • D 0 , 1 2

Q17:

Find the solution set of 2 𝑥 + 3 2 = 7 2 in ℝ .

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

Q18:

Given that 𝑦 + 1 𝑦 = 7 9 , find 𝑦 + 1 𝑦 .

  • A81
  • B9
  • C8
  • D 8 , − 8
  • E 9 , − 9

Q19:

Answer the following.

Solve 1 6 𝑥 − 2 4 𝑥 + 9 = 0 .

  • A 𝑥 = 3 4
  • B 𝑥 = 3 4 or − 3 4
  • C 𝑥 = − 3 4

Deduce from the previous question the solution to 1 6 𝑥 − 2 4 𝑥 + 9 = 0 , using a change of variable.

  • A 𝑥 = − 4 3
  • B 𝑥 = 4 3
  • C 𝑥 = 4 3 or − 4 3

Q20:

Find the solution set of 5 𝑥 + 1 2 𝑥 = − 7 in ℝ .

  • A − 7 5 , − 1
  • B 7 5 , 1
  • C − 5 7 , − 1
  • D { − 7 , − 1 }
  • E { 7 , 1 }

Q21:

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

  • A { 1 , 0 }
  • B { 1 , − 1 }
  • C { 1 }
  • D ∅

Q22:

Given that 𝑎 𝑏 = 𝑏 𝑐 = 2 , find the solution set of the equation 𝑎 𝑥 − 2 𝑏 𝑥 + 𝑐 = 0 .

  • A 1 4
  • B { 1 }
  • C 1 2
  • D { 4 }
  • E { 2 }

Q23:

Solve the equation ( 2 𝑥 − 3 ) ( 3 𝑥 + 4 ) = 0 .

  • A 𝑥 = 3 , 𝑥 = 4
  • B 𝑥 = − 3 , 𝑥 = 4
  • C 𝑥 = 3 2 , 𝑥 = − 4 3
  • D 𝑥 = − 3 2 , 𝑥 = − 4 3
  • E 𝑥 = − 3 2 , 𝑥 = 4 3

Q24:

Solve the equation 4 𝑥 + 4 0 𝑥 + 4 0 = − 6 0 by factoring.

  • A 𝑥 = 1 or 𝑥 = 2 5
  • B 𝑥 = 5
  • C 𝑥 = − 5
  • D 𝑥 = − 1 or 𝑥 = − 2 5

Q25:

Solve the equation 5 𝑥 − 1 4 𝑥 + 1 0 = 1 5 by factoring.

  • A 𝑥 = 5 7
  • B 𝑥 = 7
  • C 𝑥 = − 7
  • D 𝑥 = − 7 5
  • E 𝑥 = 7 5

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