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(𝑧−6)(𝑧−7)
  • C(𝑧+8)(𝑧+21)
  • D(𝑧−9)(𝑧−10)

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 𝑥−33𝑥+32=0.

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

Q4:

Factorize fully 24𝑥−34𝑥+12𝑥.

  • A𝑥12𝑥+42𝑥−3
  • B2𝑥3𝑥−24𝑥+3
  • C2𝑥3𝑥−24𝑥−3
  • D2𝑥3𝑥+24𝑥+3

Q5:

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

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

Q6:

Find the solution set of 𝑥−730𝑥+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 (𝑥−7)+6(𝑥−7)−72.

  • A(𝑥−19)(𝑥−1)
  • B(𝑥+3)(𝑥−24)
  • C(𝑥+5)(𝑥−13)
  • D(𝑥−3)(𝑥−25)
  • E(𝑥+12)(𝑥−6)

Q9:

Factorize fully (𝑦−4)−18(𝑦−4)+80.

  • A(𝑦+4)(𝑦+20)
  • B(𝑦−14)(𝑦−12)
  • C(𝑦+1)(𝑦+12)
  • D(𝑦−10)(𝑦−8)

Q10:

Find the solution set of 𝑥−9𝑥+14=0.

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

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.
  • B𝑥=−√3 or √3
  • C𝑥=−3 or 3
  • D𝑥=−9 or 3
  • E𝑥=−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?

  • A3𝑢−7𝑢+2=0
  • B3𝑢−7√𝑢+2=0
  • C3𝑢−7𝑢+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 3𝑥−7√𝑥+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+2𝑥−24=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 3−𝑥+6√𝑥−3−5=0.

  • A5,1
  • 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?

  • A𝑢=𝑥−5
  • B𝑢=𝑥−5
  • C𝑢=2𝑥−5
  • D𝑢=√𝑥−5
  • E𝑢=2𝑥−5

Solve for 𝑥 using a change of variable.

  • A−3√2,−3,3,3√2
  • B−√3,√3,√6,−√6
  • C−112,−1,1,112
  • D−112,−3,3,112
  • 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{−25,−1}
  • D{1,25}
  • 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{14.6,7.4}
  • C{7.3,3.7}
  • D{−14.6,7.4}

Q19:

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

  • 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 36−40×6+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 ℝ.

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

Q22:

Solve 𝑥−6𝑥+8=0loglog, where 𝑥∈ℝ.

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

Q23:

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

  • A{7}
  • B{4,3}
  • C64,1√4
  • D{64,16}
  • E16,1√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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