Worksheet: Substitution to Manipulate Equations 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)−19(𝑧+3)+90.

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

Q2:

Find the solution set of 5+1255=30.

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

Q3:

Find the solution set of 𝑥−33𝑥+32=0.

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

Q4:

Factorise fully 24𝑥−34𝑥+12𝑥.

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

Q5:

Find the solution set of 2−20×2+256=0.

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

Q6:

Find the solution set of 𝑥−730𝑥+729=0.

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

Q7:

Find the solution set of 3+7293=90.

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

Q8:

Factorise fully (𝑥−7)+6(𝑥−7)−72.

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

Q9:

Factorise fully (𝑦−4)−18(𝑦−4)+80.

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

Q10:

Find the solution set of 𝑥−9𝑥+14=0.

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

Q11:

Find the solution set of 2−10×2+32=0.

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

Q12:

Use the change of variable 𝑢=𝑥 to find the real solutions of the equation 13𝑥−2𝑥−9=0.

  • AThe equation has no solution.
  • B 𝑥 = − √ 3 or √3
  • C 𝑥 = − 3 or 3
  • D 𝑥 = − 9 or 3
  • E 𝑥 = − 3 or 9

Q13:

We want to solve 3𝑥−7√𝑥+2=0.

What equation do we get using the change of variable 𝑢=√𝑥?

  • A 3 𝑢 − 7 𝑢 + 2 = 0
  • B 3 𝑢 − 7 √ 𝑢 + 2 = 0
  • C 3 𝑢 − 7 𝑢 + 2 = 0

Solve the previous equation for 𝑢.

  • A 𝑢 = 1 9 or 4
  • B 𝑢 = − 1 3 or −2
  • C 𝑢 = − 2 3 or −1
  • D 𝑢 = 2 3 or 1
  • E 𝑢 = 1 3 or 2

Solve 3𝑥−7√𝑥+2=0.

  • A 𝑥 = 4 9 or 1
  • B 𝑥 = 1 9 or 4
  • C 𝑥 = 1 3 or 2

Q14:

Use the change of the variable 𝑢=𝑥 to find the real solutions of the equation 𝑥+2𝑥−24=0.

  • A 𝑥 = − √ 6 , 𝑥 = √ 6
  • B 𝑥 = − 6 , 𝑥 = 4
  • C 𝑥 = − 2 , 𝑥 = 2
  • D 𝑥 = − 4 , 𝑥 = 6
  • EThe equation has no solution.

Q15:

Using an appropriate change of variable, solve 3−𝑥+6√𝑥−3−5=0.

  • A 5 , 1
  • B − 5 , − 1
  • C − 2 √ 7 , − 2 , 2, 2√7

Q16:

Consider the equation 2𝑥−5+7𝑥−5−4=0.

Which of the following is the best change of variable to solve this equation?

  • A 𝑢 = 𝑥 − 5
  • B 𝑢 = 𝑥 − 5
  • C 𝑢 = 2 𝑥 − 5
  • D 𝑢 = √ 𝑥 − 5
  • E 𝑢 = 2 𝑥 − 5

Solve for 𝑥 using a change of variable.

  • A − 3 √ 2 , − 3 , 3 , 3 √ 2
  • B − √ 3 , √ 3 , √ 6 , − √ 6
  • C − 1 1 2 , − 1 , 1 , 1 1 2
  • D − 1 1 2 , − 3 , 3 , 1 1 2
  • E − 3 √ 2 , − 1 , 1 , 3 √ 2

Q17:

Find the solution set of (𝑥+4)−10(𝑥+4)+25=0 in ℝ.

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

Q18:

Find the solution set of the equation (𝑥−5)−(𝑥−5)−3=0, giving values to one decimal place.

  • A { − 7 . 3 , 3 . 7 }
  • B { 1 4 . 6 , 7 . 4 }
  • C { 7 . 3 , 3 . 7 }
  • D { − 1 4 . 6 , 7 . 4 }

Q19:

Find the solution set of 𝑥−25𝑥+144=0 in ℝ.

  • A { 8 , − 8 , 1 8 , − 1 8 }
  • B { 4 , − 4 , 3 , − 3 }
  • C { 4 , 3 }
  • D { − 1 6 , − 9 }
  • E { − 8 , − 1 8 }

Q20:

Determine the solution set of the equation 36−40×6+144=0. Give the values correct to two decimal places.

  • A { 2 . 0 0 , 0 . 7 7 }
  • B { 4 . 0 0 , 1 . 2 9 }
  • C { 6 . 0 0 , 0 . 7 7 }
  • D { 2 . 0 0 , 1 . 2 9 }
  • E { 4 . 0 0 , 0 . 3 9 }

Q21:

Find the solution set of 2+2=72 in ℝ.

  • A { − 6 , − 3 }
  • B { 6 , − 3 }
  • C { − 6 , 3 }
  • D { 6 , 3 }

Q22:

Solve 𝑥−6𝑥+8=0loglog, where 𝑥∈ℝ.

  • A { 1 0 , 2 0 }
  • B { 5 , 1 2 5 }
  • C { 2 5 , 6 2 5 }
  • D 1 2 5 , 1 6 2 5
  • E 1 1 0 , 1 2 0

Q23:

Determine the solution set of the equation loglog𝑥−1𝑥=83 in ℝ.

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

Q24:

Find the solution set of 3+243=244×3 in ℝ.

  • A { 0 , 5 }
  • B { 0 , − 5 }
  • C { 5 }
  • D { − 5 , 5 }

Q25:

Find the solution set of 4+2=516 in ℝ.

  • A { − 1 }
  • B { 3 }
  • C { 1 }
  • D { 2 }

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