Worksheet: Forming a Quadratic Equation Using Another Quadratic Equation

In this worksheet, we will practice writing a quadratic equation given the roots of another quadratic equation.

Q1:

Given that 𝐿+3 and 𝑀+3 are the roots of the equation 𝑥+8𝑥+12=0, find, in its simplest form, the quadratic equation whose roots are 𝐿 and 𝑀.

  • A𝑥+17𝑥+34=0
  • B𝑥+14𝑥+45=0
  • C𝑥17𝑥+31=0
  • D𝑥17𝑥+19=0
  • E𝑥+17𝑥+19=0

Q2:

Given that 𝐿 and 𝑀 are the roots of the equation 𝑥2𝑥+5=0, find, in its simplest form, the quadratic equation whose roots are 𝐿 and 𝑀.

  • A𝑥+14𝑥+25=0
  • B𝑥+8𝑥+25=0
  • C𝑥6𝑥+25=0
  • D𝑥+6𝑥+25=0
  • E𝑥6𝑥+10=0

Q3:

Given that 𝐿 and 𝑀 are the roots of the equation 𝑥3𝑥+12=0, find, in its simplest form, the quadratic equation whose roots are 1𝐿 and 1𝑀.

  • A𝑥15𝑥+1=0
  • B144𝑥15𝑥+1=0
  • C144𝑥+15𝑥+1=0
  • D144𝑥+5𝑥+1=0
  • E144𝑥15𝑥1=0

Q4:

Given that 𝐿 and 𝑀 are the roots of the equation 𝑥13𝑥5=0, find, in its simplest form, the quadratic equation whose roots are 𝐿+1 and 𝑀+1.

  • A𝑥15𝑥+9=0
  • B𝑥11𝑥+9=0
  • C𝑥+15𝑥+9=0
  • D𝑥15𝑥+8=0
  • E𝑥+11𝑥+8=0

Q5:

If 𝐿 and 𝑀 are the roots of the equation 𝑥+20𝑥+15=0, what is the value of 1𝑀+1𝐿?

  • A34
  • B35
  • C43
  • D43
  • E35

Q6:

Given that 𝐿 and 𝑀 are the roots of the equation 3𝑥6𝑥+7=0, find, in its simplest form, the quadratic equation whose roots are 𝐿+𝑀 and 𝐿𝑀.

  • A3𝑥13𝑥+14=0
  • B3𝑥+6𝑥+7=0
  • C3𝑥+7𝑥6=0
  • D3𝑥+13𝑥+14=0
  • E3𝑥6𝑥+7=0

Q7:

If 𝐿 and 𝑀 are the roots of the equation 𝑥19𝑥+9=0, find, in its simplest form, the quadratic equation whose roots are 𝐿2 and 𝑀2.

  • A𝑥23𝑥25=0
  • B𝑥23𝑥+32=0
  • C𝑥15𝑥25=0
  • D𝑥15𝑥+32=0
  • E𝑥+15𝑥25=0

Q8:

Given that 𝐿 and 𝑀 are the roots of the equation 𝑥+𝑥2=0, find, in its simplest form, the quadratic equation whose roots are 𝐿+𝑀 and 𝑀+𝐿.

  • A𝑥𝑥5=0
  • B𝑥+4𝑥5=0
  • C𝑥4𝑥5=0
  • D𝑥4𝑥+9=0
  • E𝑥+𝑥5=0

Q9:

Given that 𝐿 and 𝑀 are the roots of the equation 3𝑥+16𝑥1=0, find, in its simplest form, the quadratic equation whose roots are 𝐿2 and 𝑀2.

  • A12𝑥32𝑥1=0
  • B𝑥+32𝑥1=0
  • C12𝑥+32𝑥+1=0
  • D𝑥32𝑥1=0
  • E12𝑥+32𝑥1=0

Q10:

If 𝐿 and 𝑀 are the roots of the equation 2𝑥3𝑥+1=0, find, in its simplest form, the quadratic equation whose roots are 2𝐿 and 2𝑀.

  • A2𝑥5𝑥+2=0
  • B𝑥5𝑥+1=0
  • C2𝑥+5𝑥+2=0
  • D𝑥5𝑥+2=0
  • E2𝑥5𝑥+1=0

Q11:

Given that 𝐿 and 𝑀 are the roots of the equation 3𝑥+6𝑥+2=0, find, in its simplest form, the quadratic equation whose roots are 𝐿𝑀 and 𝑀𝐿.

  • A3𝑥18𝑥2=0
  • B3𝑥+6𝑥+2=0
  • C3𝑥18𝑥+2=0
  • D3𝑥+18𝑥+2=0
  • E𝑥18𝑥+2=0

Q12:

Given that 𝐿 and 𝑀 are the roots of the equation 𝑥+3𝑥5=0, find, in its simplest form, the quadratic equation whose roots are 𝐿𝑀 and 𝑀𝐿.

  • A𝑥+15𝑥+25=0
  • B𝑥15𝑥125=0
  • C𝑥+15𝑥125=0
  • D𝑥+8𝑥+25=0
  • E𝑥8𝑥125=0

Q13:

The roots of the equation 𝑥+6𝑥+𝑐=0 are 𝐿 and 𝑀, where 𝐿+𝑀=26. Find the value of 𝑐, and determine, in its simplest form, the equation whose roots are 𝐿𝑀+𝑀𝐿 and 𝐿𝑀.

  • A𝑐=10, 𝑥25𝑥+150=0
  • B𝑐=5, 𝑥+25𝑥150=0
  • C𝑐=31, 𝑥25𝑥+35=0
  • D𝑐=10, 𝑥+35𝑥+150=0
  • E𝑐=5, 𝑥25𝑥+150=0

Q14:

Given that 1𝑀 and 1𝐿 are the roots of the equation 𝑥8𝑥1=0, find, in its simplest form, the quadratic equation whose roots are 𝐿𝑀+3 and 𝐿+𝑀+6.

  • A𝑥4=0
  • B𝑥4𝑥4=0
  • C𝑥+9𝑥+36=0
  • D𝑥+4𝑥4=0
  • E𝑥+4=0

Q15:

Given that 𝐿 and 𝑀 are the roots of the equation 𝑥9𝑥7=0, find, in its simplest form, the quadratic equation whose roots are 𝐿𝑀 and 𝑀𝐿.

  • A𝑥18𝑥109=0
  • B𝑥109=0
  • C𝑥+109=0
  • D𝑥+18𝑥109=0
  • E𝑥95=0

Q16:

Given that 𝐿 and 𝑀 are the roots of the equation 𝑥16𝑥6=0, find, in its simplest form, the quadratic equation whose roots are 𝐿+𝑀 and 𝐿𝑀.

  • A𝑥16𝑥6=0
  • B𝑥+10𝑥+96=0
  • C𝑥10𝑥96=0
  • D𝑥+10𝑥96=0
  • E𝑥+16𝑥6=0

Q17:

If 𝐿 and 𝑀 are the roots of the equation 2𝑥+19𝑥11=0, find, in its simplest form, the quadratic equation whose roots are 2𝐿 and 2𝑀.

  • A2𝑥27𝑥+28=0
  • B2𝑥13𝑥+17=0
  • C𝑥+23𝑥+31=0
  • D2𝑥27𝑥+35=0
  • E𝑥+27𝑥35=0

Q18:

Given that 𝐿 and 𝑀 are the roots of the equation 2𝑥21𝑥+4=0, find, in its simplest form, the quadratic equation whose roots are 2𝐿 and 2𝑀.

  • A𝑥21𝑥+8=0
  • B𝑥+21𝑥+8=0
  • C𝑥42𝑥+16=0
  • D2𝑥21𝑥+8=0
  • E2𝑥+21𝑥+8=0

Q19:

If 𝐿 and 𝑀 are the roots of the equation 𝑥+17𝑥+1=0, find, in its simplest form, the quadratic equation whose roots are 3𝐿 and 3𝑀.

  • A𝑥34𝑥+9=0
  • B𝑥+102𝑥+9=0
  • C𝑥+51𝑥+9=0
  • D𝑥+51𝑥+3=0
  • E𝑥51𝑥+9=0

Q20:

Given that 𝐿 and 𝑀 are the roots of the equation 2𝑥10𝑥+1=0, find, in its simplest form, the quadratic equation whose roots are 𝐿3 and 𝑀3.

  • A18𝑥5𝑥+1=0
  • B18𝑥30𝑥+1=0
  • C𝑥+30𝑥+1=0
  • D18𝑥+30𝑥1=0
  • E18𝑥+30𝑥+1=0

Q21:

The roots of the equation 𝑥4𝑥+2=0 are 𝐿 and 𝑀, where 𝐿>𝑀. Find, in its simplest form, the quadratic equation whose roots are 5𝐿2𝑀 and 2𝐿5𝑀.

  • A𝑥+142𝑥+62=0
  • B𝑥142𝑥+62=0
  • C𝑥+47𝑥+82=0
  • D𝑥+22𝑥+62=0
  • E𝑥22𝑥+62=0

Q22:

Given that 𝐿 and 𝑀 are the roots of the equation 𝑥2𝑥+20=0, find, in its simplest form, the quadratic equation whose roots are 2 and 𝐿+𝑀.

  • A𝑥34𝑥36=0
  • B𝑥+34𝑥72=0
  • C𝑥34𝑥72=0
  • D𝑥36𝑥36=0
  • E𝑥36𝑥72=0

Q23:

Given that 𝑙 and 𝑚 are the roots of the equation 3𝑥+4𝑥9=0, find, in its simplest form, the quadratic equation whose roots are 𝑙 and 𝑚.

  • A9𝑥16𝑥+57=0
  • B9𝑥+70𝑥+81=0
  • C9𝑥+124𝑥+81=0
  • D9𝑥124𝑥+81=0
  • E9𝑥70𝑥+81=0

Q24:

Given that 𝐿 and 𝑀 are the roots of the equation 𝑥6𝑥7=0, find, in its simplest form, the quadratic equation whose roots are 𝐿 and 𝑀.

  • A𝑥50𝑥+49=0
  • B𝑥+𝑥+49=0
  • C𝑥+49𝑥+50=0
  • D𝑥𝑥+49=0
  • E𝑥+50𝑥+49=0

Q25:

Given that 𝐿+6 and 𝑀+6 are the roots of the equation 𝑥6𝑥4=0, find, in its simplest form, the quadratic equation whose roots are 𝐿𝑀 and 𝑀𝐿.

  • A𝑥+24𝑥+16=0
  • B𝑥+24𝑥64=0
  • C𝑥24𝑥64=0
  • D𝑥24𝑥+16=0
  • E𝑥10𝑥+24=0

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