Worksheet: Substitution to Manipulate Equations into Quadratics

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

Q1:

Factorize fully (๐‘ง+3)โˆ’19(๐‘ง+3)+90๏Šจ.

  • A(๐‘ง+3)(๐‘ง+30)
  • B(๐‘งโˆ’6)(๐‘งโˆ’7)
  • C(๐‘ง+8)(๐‘ง+21)
  • D(๐‘งโˆ’9)(๐‘งโˆ’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 ๐‘ฅโˆ’33๐‘ฅ+32=0๏Šซ๏Žค๏Žก.

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

Q4:

Factorize fully 24๐‘ฅโˆ’34๐‘ฅ+12๐‘ฅ๏Šซ๏Šฉ.

  • A๐‘ฅ๏€น12๐‘ฅ+4๏…๏€น2๐‘ฅโˆ’3๏…๏Šจ๏Šจ
  • B2๐‘ฅ๏€น3๐‘ฅโˆ’2๏…๏€น4๐‘ฅ+3๏…๏Šจ๏Šจ
  • C2๐‘ฅ๏€น3๐‘ฅโˆ’2๏…๏€น4๐‘ฅโˆ’3๏…๏Šจ๏Šจ
  • D2๐‘ฅ๏€น3๐‘ฅ+2๏…๏€น4๐‘ฅ+3๏…๏Šจ๏Šจ

Q5:

Find the solution set of 2โˆ’20ร—2+256=0๏Šจ๏—๏Šฐ๏Šจ๏—๏Šฐ๏Šจ.

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

Q6:

Find the solution set of ๐‘ฅโˆ’730๐‘ฅ+729=0๏Ž ๏Žก๏Žค๏Žฅ๏Žค.

  • A{1,243,โˆ’1,โˆ’243}
  • B{โˆ’1,โˆ’3}
  • C{โˆ’5,โˆ’15}
  • D{1,3,โˆ’1,โˆ’3}
  • E{5,15,โˆ’5,โˆ’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 (๐‘ฅโˆ’7)+6(๐‘ฅโˆ’7)โˆ’72๏Šจ.

  • A(๐‘ฅโˆ’19)(๐‘ฅโˆ’1)
  • B(๐‘ฅ+3)(๐‘ฅโˆ’24)
  • C(๐‘ฅ+5)(๐‘ฅโˆ’13)
  • D(๐‘ฅโˆ’3)(๐‘ฅโˆ’25)
  • E(๐‘ฅ+12)(๐‘ฅโˆ’6)

Q9:

Factorize fully (๐‘ฆโˆ’4)โˆ’18(๐‘ฆโˆ’4)+80๏Šจ.

  • A(๐‘ฆ+4)(๐‘ฆ+20)
  • B(๐‘ฆโˆ’14)(๐‘ฆโˆ’12)
  • C(๐‘ฆ+1)(๐‘ฆ+12)
  • D(๐‘ฆโˆ’10)(๐‘ฆโˆ’8)

Q10:

Find the solution set of ๐‘ฅโˆ’9๐‘ฅ+14=0๏Žก๏Žข๏Ž ๏Žข.

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

Q11:

Find the solution set of 2โˆ’10ร—2+32=0๏Šจ๏—๏Šฐ๏Šง๏—๏Šฐ๏Šง.

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

Q12:

Use the change of variable ๐‘ข=๐‘ฅ๏Šจ to find the real solutions of the equation 13๐‘ฅโˆ’2๐‘ฅโˆ’9=0๏Šช๏Šจ.

  • AThe equation has no solution.
  • B๐‘ฅ=โˆ’โˆš3 or โˆš3
  • C๐‘ฅ=โˆ’3 or 3
  • D๐‘ฅ=โˆ’9 or 3
  • E๐‘ฅ=โˆ’3 or 9

Q13:

We want to solve 3๐‘ฅโˆ’7โˆš๐‘ฅ+2=0.

What equation do we get using the change of variable ๐‘ข=โˆš๐‘ฅ?

  • A3๐‘ขโˆ’7๐‘ข+2=0๏Šช๏Šจ
  • B3๐‘ขโˆ’7โˆš๐‘ข+2=0
  • C3๐‘ขโˆ’7๐‘ข+2=0๏Šจ

Solve the previous equation for ๐‘ข.

  • A๐‘ข=19 or 4
  • B๐‘ข=โˆ’13 or โˆ’2
  • C๐‘ข=โˆ’23 or โˆ’1
  • D๐‘ข=23 or 1
  • E๐‘ข=13 or 2

Solve 3๐‘ฅโˆ’7โˆš๐‘ฅ+2=0.

  • A๐‘ฅ=49 or 1
  • B๐‘ฅ=19 or 4
  • C๐‘ฅ=13 or 2

Q14:

Use the change of the variable ๐‘ข=๐‘ฅ๏Šจ to find the real solutions of the equation ๐‘ฅ+2๐‘ฅโˆ’24=0๏Šช๏Šจ.

  • A๐‘ฅ=โˆ’โˆš6, ๐‘ฅ=โˆš6
  • B๐‘ฅ=โˆ’6, ๐‘ฅ=4
  • C๐‘ฅ=โˆ’2, ๐‘ฅ=2
  • D๐‘ฅ=โˆ’4, ๐‘ฅ=6
  • EThe equation has no solution.

Q15:

Using an appropriate change of variable, solve 3โˆ’๐‘ฅ+6โˆš๐‘ฅโˆ’3โˆ’5=0๏Šจ๏Šจ.

  • A5,1
  • Bโˆ’5, โˆ’1
  • Cโˆ’2โˆš7, โˆ’2, 2, 2โˆš7

Q16:

Consider the equation 2๏€น๐‘ฅโˆ’5๏…+7๏€น๐‘ฅโˆ’5๏…โˆ’4=0๏Šจ๏Šจ๏Šจ.

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

  • A๐‘ข=๏€น๐‘ฅโˆ’5๏…๏Šจ๏Šจ
  • B๐‘ข=๐‘ฅโˆ’5๏Šจ
  • C๐‘ข=2๏€น๐‘ฅโˆ’5๏…๏Šจ
  • D๐‘ข=โˆš๐‘ฅโˆ’5๏Šจ
  • E๐‘ข=2๏€น๐‘ฅโˆ’5๏…๏Šจ๏Šจ

Solve for ๐‘ฅ using a change of variable.

  • Aโˆ’3โˆš2,โˆ’3,3,3โˆš2
  • Bโˆ’โˆš3,โˆš3,โˆš6,โˆ’โˆš6
  • Cโˆ’๏„ž112,โˆ’1,1,๏„ž112
  • Dโˆ’๏„ž112,โˆ’3,3,๏„ž112
  • Eโˆ’3โˆš2,โˆ’1,1,3โˆš2

Q17:

Find the solution set of (๐‘ฅ+4)โˆ’10(๐‘ฅ+4)+25=0๏Šจ in โ„.

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

Q18:

Find the solution set of the equation (๐‘ฅโˆ’5)โˆ’(๐‘ฅโˆ’5)โˆ’3=0๏Šจ, giving values to one decimal place.

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

Q19:

Find the solution set of ๐‘ฅโˆ’25๐‘ฅ+144=0๏Šช๏Šจ in โ„.

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

Q20:

Determine the solution set of the equation 36โˆ’40ร—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๏—๏Šฏ๏Šฑ๏— in โ„.

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

Q22:

Solve ๏€บ๐‘ฅ๏†โˆ’6๐‘ฅ+8=0loglog๏Šซ๏Šจ๏Šซ, where ๐‘ฅโˆˆโ„.

  • A{10,20}
  • B{5,125}
  • C{25,625}
  • D๏ฌ125,1625๏ธ
  • E๏ฌ110,120๏ธ

Q23:

Determine the solution set of the equation loglog๏Šช๏Šช๐‘ฅโˆ’1๐‘ฅ=83 in โ„.

  • A{7}
  • B{4,3}
  • C๏ฏ64,1โˆš4๏ป๏Žข
  • D{64,16}
  • E๏ฏ16,1โˆš2๏ป

Q24:

Find the solution set of 3+243=244ร—3๏Šจ๏—๏— in โ„.

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

Q25:

Find the solution set of 4+2=516๏—๏—๏Šฐ๏Šฎ in โ„.

  • A{โˆ’1}
  • B{3}
  • C{1}
  • D{2}

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