Lesson Worksheet: Substitution to Manipulate Equations into Quadratics Mathematics

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

Q1:

Factorize fully (𝑧+3)19(𝑧+3)+90.

  • A(𝑧+3)(𝑧+30)
  • B(𝑧6)(𝑧7)
  • C(𝑧+8)(𝑧+21)
  • D(𝑧9)(𝑧10)

Q2:

Find the solution set of 5+1255=30.

  • A{1,2}
  • B{4,44}
  • C{5,25}
  • D{10,100}

Q3:

Find the solution set of 𝑥33𝑥+32=0.

  • A{1,2}
  • B{1,4}
  • C{1,32}
  • D{4,16}
  • E{2}

Q4:

Factorize fully 24𝑥34𝑥+12𝑥.

  • A𝑥12𝑥+42𝑥3
  • B2𝑥3𝑥24𝑥+3
  • C2𝑥3𝑥24𝑥3
  • D2𝑥3𝑥+24𝑥+3

Q5:

Find the solution set of 220×2+256=0.

  • A{6,6}
  • B{2,4}
  • C{3,3}
  • D{4,16}
  • E{2,2}

Q6:

Find the solution set of 𝑥730𝑥+729=0.

  • A{1,243,1,243}
  • B{1,3}
  • C{5,15}
  • D{1,3,1,3}
  • E{5,15,5,15}

Q7:

Find the solution set of 3+7293=90.

  • A{2,4}
  • B{8,84}
  • C{9,81}
  • D{36,360}

Q8:

Factorize fully (𝑥7)+6(𝑥7)72.

  • A(𝑥19)(𝑥1)
  • B(𝑥+3)(𝑥24)
  • C(𝑥+5)(𝑥13)
  • D(𝑥3)(𝑥25)
  • E(𝑥+12)(𝑥6)

Q9:

Factorize fully (𝑦4)18(𝑦4)+80.

  • A(𝑦+4)(𝑦+20)
  • B(𝑦14)(𝑦12)
  • C(𝑦+1)(𝑦+12)
  • D(𝑦10)(𝑦8)

Q10:

Find the solution set of 𝑥9𝑥+14=0.

  • A{343,8}
  • B{7,2}
  • C{21,6}
  • D{7,2}
  • E{21,6}

Q11:

Find the solution set of 210×2+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 13𝑥2𝑥9=0.

  • AThe equation has no solution.
  • B𝑥=3 or 3
  • C𝑥=3 or 3
  • D𝑥=9 or 3
  • E𝑥=3 or 9

Q13:

We want to solve 3𝑥7𝑥+2=0.

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

  • A3𝑢7𝑢+2=0
  • B3𝑢7𝑢+2=0
  • C3𝑢7𝑢+2=0

Solve the previous equation for 𝑢.

  • A𝑢=19 or 4
  • B𝑢=13 or 2
  • C𝑢=23 or 1
  • D𝑢=23 or 1
  • E𝑢=13 or 2

Solve 3𝑥7𝑥+2=0.

  • A𝑥=49 or 1
  • B𝑥=19 or 4
  • C𝑥=13 or 2

Q14:

Use the change of the variable 𝑢=𝑥 to find the real solutions of the equation 𝑥+2𝑥24=0.

  • A𝑥=6, 𝑥=6
  • B𝑥=6, 𝑥=4
  • C𝑥=2, 𝑥=2
  • D𝑥=4, 𝑥=6
  • EThe equation has no solution.

Q15:

Using an appropriate change of variable, solve 3𝑥+6𝑥35=0.

  • A5,1
  • B5, 1
  • C27, 2, 2, 27

Q16:

Consider the equation 2𝑥5+7𝑥54=0.

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

  • A𝑢=𝑥5
  • B𝑢=𝑥5
  • C𝑢=2𝑥5
  • D𝑢=𝑥5
  • E𝑢=2𝑥5

Solve for 𝑥 using a change of variable.

  • A32,3,3,32
  • B3,3,6,6
  • C112,1,1,112
  • D112,3,3,112
  • E32,1,1,32

Q17:

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

  • A{1}
  • B{5}
  • C{25,1}
  • D{1,25}
  • E{1}

Q18:

Find the solution set of the equation (𝑥5)(𝑥5)3=0, giving values to one decimal place.

  • A{7.3,3.7}
  • B{14.6,7.4}
  • C{7.3,3.7}
  • D{14.6,7.4}

Q19:

Find the solution set of 𝑥25𝑥+144=0 in .

  • A{8,8,18,18}
  • B{4,4,3,3}
  • C{4,3}
  • D{16,9}
  • E{8,18}

Q20:

Determine the solution set of the equation 3640×6+144=0. Give the values correct to two decimal places.

  • A{2.00,0.77}
  • B{4.00,1.29}
  • C{6.00,0.77}
  • D{2.00,1.29}
  • E{4.00,0.39}

Q21:

Find the solution set of 2+2=72 in .

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

Q22:

Solve 𝑥6𝑥+8=0loglog, where 𝑥.

  • A{10,20}
  • B{5,125}
  • C{25,625}
  • D125,1625
  • E110,120

Q23:

Determine the solution set of the equation loglog𝑥1𝑥=83 in .

  • A{7}
  • B{4,3}
  • C64,14
  • D{64,16}
  • E16,12

Q24:

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

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

Q25:

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

  • A{1}
  • B{3}
  • C{1}
  • D{2}

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