Worksheet: Writing a Quadratic Equation given the Roots

In this worksheet, we will practice writing a quadratic equation given its roots.

Q1:

Find, in its simplest form, the quadratic equation whose roots are 8 1 1 and 1 1 .

  • A 𝑥 7 𝑥 8 8 = 0
  • B 𝑥 + 7 1 1 𝑥 8 8 = 0
  • C 𝑥 7 1 1 𝑥 + 8 8 = 0
  • D 𝑥 7 1 1 𝑥 8 8 = 0
  • E 8 8 𝑥 7 1 1 𝑥 8 8 = 0

Q2:

What is the sum of the roots of the equation 𝑥 2 4 = 0 ?

Q3:

If the roots of the equation 5 𝑥 2 𝑘 𝑥 + 5 = 0 are equal, what are the possible values of 𝑘 ?

  • A 1 0 , 1 0
  • B0
  • C10
  • D 5 , 5
  • E 1 0

Q4:

What is the simplest form of the quadratic equation whose roots are 1 3 2 and 5 3 ?

  • A 6 𝑥 + 2 9 𝑥 6 5 = 0
  • B 6 𝑥 + 4 9 𝑥 + 6 5 = 0
  • C 2 𝑥 3 𝑥 6 5 = 0
  • D 6 𝑥 4 9 𝑥 + 6 5 = 0

Q5:

Find, in its simplest form, the quadratic equation whose roots are 𝑚 + 3 𝑛 and 𝑚 3 𝑛 .

  • A 𝑥 2 𝑥 + 𝑚 + 9 𝑛 = 0
  • B 𝑥 2 𝑚 𝑥 + 𝑚 3 𝑛 = 0
  • C 𝑥 + 2 𝑥 + 𝑚 9 𝑛 = 0
  • D 𝑥 2 𝑚 𝑥 + 𝑚 9 𝑛 = 0
  • E 𝑥 6 𝑛 𝑥 + 𝑚 9 𝑛 = 0

Q6:

Find, in its simplest form, the quadratic equation whose roots are 4 2 2 𝑖 5 3 𝑖 and 4 + 4 6 𝑖 4 5 𝑖 .

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

Q7:

Given that 1 and 6 are the solutions of the equation 𝑥 + 𝑏 𝑥 + 𝑐 = 0 , find the values of 𝑏 and 𝑐 .

  • A 𝑏 = 6 , 𝑐 = 7
  • B 𝑏 = 7 , 𝑐 = 6
  • C 𝑏 = 6 , 𝑐 = 7
  • D 𝑏 = 7 , 𝑐 = 6
  • E 𝑏 = 7 , 𝑐 = 6

Q8:

Given that 𝑥 = 9 is a root of the equation 𝑥 + 𝑚 𝑥 = 3 6 , determine the value of 𝑚 .

Q9:

If one of the roots of the equation 3 𝑥 + 9 𝑥 = 0 is a root of the equation 𝑥 + 1 2 𝑥 + 𝑎 = 0 , what is the value of 𝑎 ?

  • A 2 7
  • B 2 7 or 0
  • C0
  • D27 or 0

Q10:

The sum, 𝑆 , of the first 𝑛 consecutive integers ( 1 + 2 + 3 + 4 + + 𝑛 ) can be found using 𝑆 = 𝑛 2 ( 1 + 𝑛 ) . Starting from 1, how many consecutive integers are required to make a sum of 1,953?

Q11:

Find the solution set of 𝑥 5 𝑥 + 1 2 5 = 1 8 5 in .

  • A { 1 , 6 }
  • B { 2 , 3 }
  • C { 1 , 6 }
  • D { 2 , 3 }

Q12:

Find the positive number whose square is 12 more than four times the number.

Q13:

If 𝐿 and 𝑀 are the roots of the equation 𝑥 + 1 0 𝑥 + 9 = 0 , what is the value of 𝐿 + 𝑀 ?

Q14:

Given that 1 is one of the roots of the equation 𝑥 + 𝑎 𝑥 + 2 = 0 , find the value of 𝑎 and the value of the other root.

  • A 𝑎 = 3 , other root = 2
  • B 𝑎 = 3 , other root = 2
  • C 𝑎 = 3 , other root = 2
  • D 𝑎 = 3 , other root = 2

Q15:

If the two roots of the equation 5 𝑥 + 1 0 𝑥 + 𝑘 = 0 are equal, determine the value of 𝑘 , then find the two roots.

  • A 𝑘 = 5 , roots: 1 , 1
  • B 𝑘 = 5 , roots: 1, 1
  • C 𝑘 = 5 , roots: 1 , 1
  • D 𝑘 = 5 , roots: 1, 1

Q16:

Given that a root of the equation 𝑥 + 1 8 𝑥 + 𝑘 = 0 is 𝑥 = 3 , what is the value of 𝑘 ?

Q17:

If 𝐿 and 𝑀 are the roots of the equation 𝑥 + 2 𝑥 6 = 0 , what is the value of 𝐿 + 𝑀 ?

Q18:

Find the values of 𝑐 for which one of the roots of the equation 6 𝑥 7 2 𝑥 + 𝑐 = 0 is the square of the other.

  • A 2 4 , 18
  • B384, 1 6 2
  • C24, 1 8
  • D 3 8 4 , 162
  • E 6 4 , 27

Q19:

Given that 1 and 12 are the roots of the equation 𝑥 + 𝑚 𝑥 + 𝑛 = 0 , find the values of 𝑚 and 𝑛 .

  • A 𝑚 = 1 2 , 𝑛 = 1 3
  • B 𝑚 = 1 3 , 𝑛 = 1 2
  • C 𝑚 = 1 1 , 𝑛 = 1 3
  • D 𝑚 = 1 3 , 𝑛 = 1 2
  • E 𝑚 = 1 2 , 𝑛 = 1 3

Q20:

Given that 𝑥 = 5 + 6 𝑖 is one of the roots of the equation 4 𝑥 + 𝑘 𝑥 2 4 4 = 0 , find the other root and the value of 𝑘 .

  • A 𝑥 = 4 8 , 𝑘 = 1 0
  • B 𝑥 = 5 6 𝑖 , 𝑘 = 4 0
  • C 𝑥 = 4 8 , 𝑘 = 4 0
  • D 𝑥 = 5 6 𝑖 , 𝑘 = 4 0
  • E 𝑥 = 5 6 𝑖 , 𝑘 = 1 0

Q21:

Given that 𝑥 = 2 is a solution of the equation 𝑥 + 𝑏 𝑥 2 4 = 0 , find the value of 𝑏 .

Q22:

Given that 3 is a solution of the equation 9 𝑥 + 7 𝑥 + 𝑘 = 0 , find the value of 𝑘 .

Q23:

Without solving the equation 3 𝑥 1 6 𝑥 + 6 3 = 0 , find the sum of its roots.

  • A 3 1 6
  • B 3 2 3
  • C 1 6 3 9 𝑖
  • D 1 6 3
  • E 2 1

Q24:

Given that 𝐿 and 𝐿 are the roots of the equation 4 𝑥 + 𝑏 𝑥 + 3 2 = 0 , find the value of 𝑏 .

Q25:

Given that the sum of the roots of the equation 8 𝑥 + 𝑏 𝑥 + 1 8 = 0 is equal to their product, find the value of 𝑏 .

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