Worksheet: Forming a Quadratic Equation from its 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 811 and 11.

  • A 𝑥 7 1 1 𝑥 8 8 = 0
  • B 𝑥 + 7 1 1 𝑥 8 8 = 0
  • C 𝑥 7 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 𝑥24=0?

Q3:

What is the simplest form of the quadratic equation whose roots are 132 and 53?

  • 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

Q4:

Find, in its simplest form, the quadratic equation whose roots are 𝑚+3𝑛 and 𝑚3𝑛.

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

Q5:

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

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

Q6:

Given that 𝑥=9 is a root of the equation 𝑥+𝑚𝑥=36, determine the value of 𝑚.

Q7:

If one of the roots of the equation 3𝑥+9𝑥=0 is a root of the equation 𝑥+12𝑥+𝑎=0, what is the value of 𝑎?

  • A 2 7 or 0
  • B 2 7
  • C27 or 0
  • D0

Q8:

If 𝐿 and 𝑀 are the roots of the equation 𝑥+10𝑥+9=0, what is the value of 𝐿+𝑀?

Q9:

Given that 1 is one of the roots of the equation 𝑥+𝑎𝑥+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

Q10:

Given that a root of the equation 𝑥+18𝑥+𝑘=0 is 𝑥=3, what is the value of 𝑘?

Q11:

If 𝐿 and 𝑀 are the roots of the equation 𝑥+2𝑥6=0, what is the value of 𝐿+𝑀?

Q12:

Find the values of 𝑐 for which one of the roots of the equation 6𝑥72𝑥+𝑐=0 is the square of the other.

  • A24, 18
  • B 2 4 , 18
  • C 6 4 , 27
  • D 3 8 4 , 162
  • E384, 162

Q13:

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

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

Q14:

Given that 𝑥=2 is a solution of the equation 𝑥+𝑏𝑥24=0, find the value of 𝑏.

Q15:

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

Q16:

Without solving the equation 3𝑥16𝑥+63=0, find the sum of its roots.

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

Q17:

Given that 𝐿 and 𝐿 are the roots of the equation 4𝑥+𝑏𝑥+32=0, find the value of 𝑏.

Q18:

Given that the sum of the roots of the equation 8𝑥+𝑏𝑥+18=0 is equal to their product, find the value of 𝑏.

Q19:

What is the product of the roots of the quadratic equation 𝑎𝑥+𝑏𝑥+𝑐=0?

  • A 𝑏 𝑐
  • B 𝑏 𝑎
  • C 𝑐 𝑎
  • D 𝑎 𝑏

Q20:

If the sum of the roots of the equation 3𝑥+𝑘𝑥+11=0 is 4, what is the value of 𝑘?

Q21:

Given that 𝑙 and 4𝑙 are the roots of the equation 𝑥𝑚𝑥17=0, find the value of 𝑚.

Q22:

The difference between the roots of the equation 9𝑥4𝑥2=𝑐 is 49. What is the value of 𝑐?

Q23:

The roots of the equation 𝑚𝑥12𝑛𝑥+𝑙=0, where 𝑚0 are 𝐿 and 𝑀. Given that 𝐿>𝑀 and 𝐿𝑀=20, does 𝐿=10+6𝑛𝑚?

  • Ano
  • Byes

Q24:

If the product of the roots of the equation 6𝑥+2𝑥+𝑘=0 is 4, what is the value of 𝑘?

Q25:

The roots of the equation 6𝑥𝑚𝑥+24=0 are positive numbers which are in the ratio 49. What is the value of 𝑚?

  • A 1 3 3
  • B72
  • C26
  • D 2 6 3

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