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.

  • A 𝑥 + 1 7 𝑥 + 3 4 = 0
  • B 𝑥 + 1 4 𝑥 + 4 5 = 0
  • C 𝑥 1 7 𝑥 + 3 1 = 0
  • D 𝑥 1 7 𝑥 + 1 9 = 0
  • E 𝑥 + 1 7 𝑥 + 1 9 = 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.

  • A 𝑥 + 1 4 𝑥 + 2 5 = 0
  • B 𝑥 + 8 𝑥 + 2 5 = 0
  • C 𝑥 6 𝑥 + 2 5 = 0
  • D 𝑥 + 6 𝑥 + 2 5 = 0
  • E 𝑥 6 𝑥 + 1 0 = 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.

  • A 𝑥 1 5 𝑥 + 1 = 0
  • B 1 4 4 𝑥 1 5 𝑥 + 1 = 0
  • C 1 4 4 𝑥 + 1 5 𝑥 + 1 = 0
  • D 1 4 4 𝑥 + 5 𝑥 + 1 = 0
  • E 1 4 4 𝑥 1 5 𝑥 1 = 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.

  • A 𝑥 1 5 𝑥 + 9 = 0
  • B 𝑥 1 1 𝑥 + 9 = 0
  • C 𝑥 + 1 5 𝑥 + 9 = 0
  • D 𝑥 1 5 𝑥 + 8 = 0
  • E 𝑥 + 1 1 𝑥 + 8 = 0

Q5:

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

  • A 3 4
  • B 3 5
  • C 4 3
  • D 4 3
  • 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.

  • A 3 𝑥 1 3 𝑥 + 1 4 = 0
  • B 3 𝑥 + 6 𝑥 + 7 = 0
  • C 3 𝑥 + 7 𝑥 6 = 0
  • D 3 𝑥 + 1 3 𝑥 + 1 4 = 0
  • E 3 𝑥 6 𝑥 + 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.

  • A 𝑥 2 3 𝑥 2 5 = 0
  • B 𝑥 2 3 𝑥 + 3 2 = 0
  • C 𝑥 1 5 𝑥 2 5 = 0
  • D 𝑥 1 5 𝑥 + 3 2 = 0
  • E 𝑥 + 1 5 𝑥 2 5 = 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.

  • A 𝑥 𝑥 5 = 0
  • B 𝑥 + 4 𝑥 5 = 0
  • C 𝑥 4 𝑥 5 = 0
  • D 𝑥 4 𝑥 + 9 = 0
  • E 𝑥 + 𝑥 5 = 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.

  • A 1 2 𝑥 3 2 𝑥 1 = 0
  • B 𝑥 + 3 2 𝑥 1 = 0
  • C 1 2 𝑥 + 3 2 𝑥 + 1 = 0
  • D 𝑥 3 2 𝑥 1 = 0
  • E 1 2 𝑥 + 3 2 𝑥 1 = 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.

  • A 2 𝑥 5 𝑥 + 2 = 0
  • B 𝑥 5 𝑥 + 1 = 0
  • C 2 𝑥 + 5 𝑥 + 2 = 0
  • D 𝑥 5 𝑥 + 2 = 0
  • E 2 𝑥 5 𝑥 + 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.

  • A 3 𝑥 1 8 𝑥 + 2 = 0
  • B 3 𝑥 + 6 𝑥 + 2 = 0
  • C 𝑥 1 8 𝑥 + 2 = 0
  • D 3 𝑥 1 8 𝑥 2 = 0
  • E 3 𝑥 + 1 8 𝑥 + 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.

  • A 𝑥 + 1 5 𝑥 + 2 5 = 0
  • B 𝑥 1 5 𝑥 1 2 5 = 0
  • C 𝑥 + 1 5 𝑥 1 2 5 = 0
  • D 𝑥 + 8 𝑥 + 2 5 = 0
  • E 𝑥 8 𝑥 1 2 5 = 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.

  • A 𝑐 = 1 0 , 𝑥 2 5 𝑥 + 1 5 0 = 0
  • B 𝑐 = 5 , 𝑥 + 2 5 𝑥 1 5 0 = 0
  • C 𝑐 = 3 1 , 𝑥 2 5 𝑥 + 3 5 = 0
  • D 𝑐 = 1 0 , 𝑥 + 3 5 𝑥 + 1 5 0 = 0
  • E 𝑐 = 5 , 𝑥 2 5 𝑥 + 1 5 0 = 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.

  • A 𝑥 4 = 0
  • B 𝑥 4 𝑥 4 = 0
  • C 𝑥 + 9 𝑥 + 3 6 = 0
  • D 𝑥 + 4 𝑥 4 = 0
  • E 𝑥 + 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.

  • A 𝑥 1 8 𝑥 1 0 9 = 0
  • B 𝑥 1 0 9 = 0
  • C 𝑥 + 1 0 9 = 0
  • D 𝑥 + 1 8 𝑥 1 0 9 = 0
  • E 𝑥 9 5 = 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.

  • A 𝑥 1 6 𝑥 6 = 0
  • B 𝑥 + 1 0 𝑥 + 9 6 = 0
  • C 𝑥 1 0 𝑥 9 6 = 0
  • D 𝑥 + 1 0 𝑥 9 6 = 0
  • E 𝑥 + 1 6 𝑥 6 = 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.

  • A 2 𝑥 2 7 𝑥 + 2 8 = 0
  • B 2 𝑥 1 3 𝑥 + 1 7 = 0
  • C 𝑥 + 2 3 𝑥 + 3 1 = 0
  • D 2 𝑥 2 7 𝑥 + 3 5 = 0
  • E 𝑥 + 2 7 𝑥 3 5 = 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.

  • A 𝑥 2 1 𝑥 + 8 = 0
  • B 𝑥 + 2 1 𝑥 + 8 = 0
  • C 𝑥 4 2 𝑥 + 1 6 = 0
  • D 2 𝑥 2 1 𝑥 + 8 = 0
  • E 2 𝑥 + 2 1 𝑥 + 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.

  • A 𝑥 3 4 𝑥 + 9 = 0
  • B 𝑥 + 1 0 2 𝑥 + 9 = 0
  • C 𝑥 + 5 1 𝑥 + 9 = 0
  • D 𝑥 + 5 1 𝑥 + 3 = 0
  • E 𝑥 5 1 𝑥 + 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.

  • A 1 8 𝑥 5 𝑥 + 1 = 0
  • B 1 8 𝑥 3 0 𝑥 + 1 = 0
  • C 𝑥 + 3 0 𝑥 + 1 = 0
  • D 1 8 𝑥 + 3 0 𝑥 1 = 0
  • E 1 8 𝑥 + 3 0 𝑥 + 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.

  • A 𝑥 + 1 4 2 𝑥 + 6 2 = 0
  • B 𝑥 1 4 2 𝑥 + 6 2 = 0
  • C 𝑥 + 4 7 𝑥 + 8 2 = 0
  • D 𝑥 + 2 2 𝑥 + 6 2 = 0
  • E 𝑥 2 2 𝑥 + 6 2 = 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.

  • A 𝑥 3 4 𝑥 3 6 = 0
  • B 𝑥 + 3 4 𝑥 7 2 = 0
  • C 𝑥 3 4 𝑥 7 2 = 0
  • D 𝑥 3 6 𝑥 3 6 = 0
  • E 𝑥 3 6 𝑥 7 2 = 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.

  • A 9 𝑥 1 6 𝑥 + 5 7 = 0
  • B 9 𝑥 + 7 0 𝑥 + 8 1 = 0
  • C 9 𝑥 + 1 2 4 𝑥 + 8 1 = 0
  • D 9 𝑥 1 2 4 𝑥 + 8 1 = 0
  • E 9 𝑥 7 0 𝑥 + 8 1 = 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.

  • A 𝑥 5 0 𝑥 + 4 9 = 0
  • B 𝑥 + 𝑥 + 4 9 = 0
  • C 𝑥 + 4 9 𝑥 + 5 0 = 0
  • D 𝑥 𝑥 + 4 9 = 0
  • E 𝑥 + 5 0 𝑥 + 4 9 = 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.

  • A 𝑥 + 2 4 𝑥 + 1 6 = 0
  • B 𝑥 + 2 4 𝑥 6 4 = 0
  • C 𝑥 2 4 𝑥 6 4 = 0
  • D 𝑥 2 4 𝑥 + 1 6 = 0
  • E 𝑥 1 0 𝑥 + 2 4 = 0

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