Worksheet: Writing a Quadratic Equation given the Roots of Another Quadratic Equation

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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 + 8 𝑥 + 1 2 = 0 , find, in its simplest form, the quadratic equation whose roots are 𝐿 and 𝑀 .

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

Q2:

Given that 𝐿 and 𝑀 are the roots of the equation 𝑥 − 2 𝑥 + 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 − 6 𝑥 + 2 5 = 0
  • C 𝑥 − 6 𝑥 + 1 0 = 0
  • D 𝑥 + 6 𝑥 + 2 5 = 0
  • E 𝑥 + 8 𝑥 + 2 5 = 0

Q3:

Given that 𝐿 and 𝑀 are the roots of the equation 𝑥 − 3 𝑥 + 1 2 = 0 , find, in its simplest form, the quadratic equation whose roots are 1 𝐿 and 1 𝑀 .

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

Q4:

Given that 𝐿 and 𝑀 are the roots of the equation 𝑥 − 1 3 𝑥 − 5 = 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 + 8 = 0
  • D 𝑥 − 1 5 𝑥 + 9 = 0
  • E 𝑥 + 1 1 𝑥 + 8 = 0

Q5:

If 𝐿 and 𝑀 are the roots of the equation 𝑥 + 2 0 𝑥 + 1 5 = 0 , what is the value of 1 𝑀 + 1 𝐿 ?

  • A − 3 4
  • B 4 3
  • C − 3 5
  • D − 4 3
  • E35

Q6:

Without solving the equation 3 𝑥 − 3 𝑥 − 2 = 𝑥 + 5 𝑥 − 7 , find the sum and the product of its roots.

  • AThe sum is − 2 7 , and the product is 31.
  • BThe sum is − 2 7 2 , and the product is 3 1 2 .
  • CThe sum is 27, and the product is 31.
  • DThe sum is 2 7 2 , and the product is 3 1 2 .
  • EThe sum is − 2 7 , and the product is 2.

Q7:

Given that 𝐿 and 𝑀 are the roots of the equation 3 𝑥 − 6 𝑥 + 7 = 0 , find, in its simplest form, the quadratic equation whose roots are 𝐿 + 𝑀 and 𝐿 𝑀 .

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

Q8:

The roots of the equation 𝑚 𝑥 + 6 𝑛 𝑥 + 𝑙 = 0 , where 𝑚 ≠ 0 , are 𝐿 and 𝑀 . Given that 𝐿 − 𝑀 = 2 1 𝐿 + 1 𝑀 , does 𝑙 9 𝑛 − 𝑚 𝑙 = 3 6 𝑛 𝑚 ?

  • Ayes
  • Bno

Q9:

Without solving the equation ( 7 𝑥 + 2 ) ( 8 𝑥 + 1 ) = 0 , find the sum of its roots.

  • A − 5 6 2 3
  • B − 2 3 2 8
  • C − 2 3 5 6 + 5 1 4 𝑖
  • D − 2 3 5 6
  • E − 2 3 5 6 + 9 5 6 𝑖

Q10:

What is the product of the roots of the equation 𝑚 𝑥 + 4 𝑛 𝑥 + 8 𝑙 = 0 ?

  • A 4 𝑛 𝑚
  • B 8 𝑙
  • C − 4 𝑛 𝑚
  • D 8 𝑙 𝑚
  • E − 4 𝑛

Q11:

Given that √ 3 𝑖 and − √ 3 𝑖 are the two roots of the equation 𝑥 + 𝑚 𝑥 + 𝑛 = 0 , find the values of 𝑚 and 𝑛 .

  • A 𝑚 = 0 , 𝑛 = 9
  • B 𝑚 = 0 , 𝑛 = − 3
  • C 𝑚 = 0 , 𝑛 = − 9
  • D 𝑚 = 0 , 𝑛 = 3
  • E 𝑚 = 3 , 𝑛 = 0

Q12:

Find the sum and the product of the roots of the equation ( 4 𝑥 + 1 ) ( 𝑥 + 7 ) = ( 𝑥 + 4 ) ( 𝑥 − 8 ) without solving it.

  • AThe sum is 33, the product is 39.
  • BThe sum is 11, the product is 13.
  • CThe sum is − 3 3 , the product is 39.
  • DThe sum is − 1 1 , the product is 13.
  • EThe sum is 33, the product is 3.

Q13:

If 𝐿 and 𝑀 are the roots of the equation 𝑥 + 2 2 𝑥 − 1 2 = 0 , what is the value of 𝐿 + 1 𝑀 𝑀 + 1 𝐿 ?

  • A − 1 1 1 1 2
  • B − 1 0
  • C 1 5 6
  • D − 1 0 1 1 2
  • E10

Q14:

Given that each root of the equation 5 𝑥 + 𝑘 = 𝑥 + 4 is the multiplicative inverse of the other, find all possible values of 𝑘 .

  • A3
  • B9
  • C 9 , − 9
  • D 3 , − 3
  • E1

Q15:

The sum of the roots of the equation 𝑥 − ( 𝑘 + 6 ) 𝑥 − 6 𝑘 = 0 is equal to the product of the roots of the equation 3 𝑥 + 9 𝑘 𝑥 + 𝑘 = 0 . Find the possible values of 𝑘 .

  • A1 or − 1 8
  • B3 or − 6
  • C − 1 or − 1 8
  • D − 3 or 6
  • E3 or 12

Q16:

The product of the roots of the equation 2 𝑥 − 4 𝑥 + 6 𝑘 = 0 is equal to the sum of the roots of the equation 𝑥 − ( 𝑘 − 8 ) 𝑥 = 0 . Find the value of 𝑘 .

Q17:

If 𝐿 and 𝑀 are the roots of the equation 𝑥 − 1 9 𝑥 + 9 = 0 , find, in its simplest form, the quadratic equation whose roots are 𝐿 − 2 and 𝑀 − 2 .

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

Q18:

Given that 𝐿 and 𝑀 are the roots of the equation 𝑥 + 𝑥 − 2 = 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 − 𝑥 − 5 = 0
  • D 𝑥 − 4 𝑥 − 5 = 0
  • E 𝑥 − 4 𝑥 + 9 = 0

Q19:

Given that 𝐿 and 𝑀 are the roots of the equation 3 𝑥 + 1 6 𝑥 − 1 = 0 , find, in its simplest form, the quadratic equation whose roots are 𝐿 2 and 𝑀 2 .

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

Q20:

If 𝐿 and 𝑀 are the roots of the equation 2 𝑥 − 3 𝑥 + 1 = 0 , find, in its simplest form, the quadratic equation whose roots are 2 𝐿 and 2 𝑀 .

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

Q21:

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

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

Q22:

Given that 𝐿 and 𝑀 are the roots of the equation 𝑥 + 3 𝑥 − 5 = 0 , find, in its simplest form, the quadratic equation whose roots are 𝐿 𝑀 and 𝑀 𝐿 .

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

Q23:

The roots of the equation 𝑥 + 6 𝑥 + 𝑐 = 0 are 𝐿 and 𝑀 , where 𝐿 + 𝑀 = 2 6 . Find the value of 𝑐 , and determine, in its simplest form, the equation whose roots are 𝐿 𝑀 + 𝑀 𝐿 and 𝐿 𝑀 .

  • 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 = − 1 0 , 𝑥 + 3 5 𝑥 + 1 5 0 = 0
  • D 𝑐 = 5 , 𝑥 + 2 5 𝑥 − 1 5 0 = 0
  • E 𝑐 = 3 1 , 𝑥 − 2 5 𝑥 + 3 5 = 0

Q24:

Given that 1 𝑀 and 1 𝐿 are the roots of the equation 𝑥 − 8 𝑥 − 1 = 0 , find, in its simplest form, the quadratic equation whose roots are 𝐿 𝑀 + 3 and 𝐿 + 𝑀 + 6 .

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

Q25:

Given that 𝐿 and 𝑀 are the roots of the equation 𝑥 − 9 𝑥 − 7 = 0 , find, in its simplest form, the quadratic equation whose roots are 𝐿 − 𝑀 and 𝑀 − 𝐿 .

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

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