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Lesson: Writing a Quadratic Equation given the Roots

In this lesson, we will learn how to write a quadratic equation given its roots.

Sample Question Videos

01:11

Worksheet • 25 Questions • 1 Video

Q1:

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

  • A π‘₯ βˆ’ 7 √ 1 1 π‘₯ βˆ’ 8 8 = 0 2
  • B βˆ’ 8 8 π‘₯ βˆ’ 7 √ 1 1 π‘₯ βˆ’ 8 8 = 0 2
  • C π‘₯ + 7 √ 1 1 π‘₯ βˆ’ 8 8 = 0 2
  • D π‘₯ βˆ’ 7 √ 1 1 π‘₯ + 8 8 = 0 2
  • E π‘₯ βˆ’ 7 π‘₯ βˆ’ 8 8 = 0 2

Q2:

Find the values of 𝑐 for which one of the roots of the equation 6 π‘₯ βˆ’ 7 2 π‘₯ + 𝑐 = 0 2 is the square of the other.

  • A βˆ’ 3 8 4 , 162
  • B βˆ’ 6 4 , 27
  • C384, βˆ’ 1 6 2
  • D24, βˆ’ 1 8
  • E βˆ’ 2 4 , 18

Q3:

Given that βˆ’ 1 and βˆ’ 6 are the solutions of the equation π‘₯ + 𝑏 π‘₯ + 𝑐 = 0 2 , find the values of 𝑏 and 𝑐 .

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

Q4:

Given that π‘₯ = βˆ’ 9 is a root of the equation π‘₯ + π‘š π‘₯ = 3 6 2 , determine the value of π‘š .

Q5:

If one of the roots of the equation 3 π‘₯ + 9 π‘₯ = 0 2 is a root of the equation π‘₯ + 1 2 π‘₯ + π‘Ž = 0 2 , what is the value of π‘Ž ?

  • A27 or 0
  • B βˆ’ 2 7 or 0
  • C0
  • D βˆ’ 2 7

Q6:

Given that 1 and 12 are the roots of the equation π‘₯ + π‘š π‘₯ + 𝑛 = 0 2 , find the values of π‘š and 𝑛 .

  • A π‘š = βˆ’ 1 3 , 𝑛 = 1 2
  • B π‘š = 1 2 , 𝑛 = 1 3
  • C π‘š = 1 3 , 𝑛 = 1 2
  • D π‘š = βˆ’ 1 1 , 𝑛 = 1 3
  • E π‘š = 1 2 , 𝑛 = βˆ’ 1 3

Q7:

Given that π‘₯ = 5 + 6 𝑖 is one of the roots of the equation βˆ’ 4 π‘₯ + π‘˜ π‘₯ βˆ’ 2 4 4 = 0 2 , find the other root and the value of π‘˜ .

  • A π‘₯ = 5 βˆ’ 6 𝑖 , π‘˜ = 4 0
  • B π‘₯ = 5 βˆ’ 6 𝑖 , π‘˜ = 1 0
  • C π‘₯ = 5 βˆ’ 6 𝑖 , π‘˜ = βˆ’ 4 0
  • D π‘₯ = βˆ’ 4 8 , π‘˜ = βˆ’ 4 0
  • E π‘₯ = βˆ’ 4 8 , π‘˜ = 1 0

Q8:

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

Q9:

Find the solution set of π‘₯ βˆ’ 5 π‘₯ + 1 2 5 = 1 8 5 2 in ℝ .

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

Q10:

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

Q11:

If 𝐿 and 𝑀 are the roots of the equation π‘₯ + 1 0 π‘₯ + 9 = 0 2 , what is the value of 𝐿 + 𝑀 2 2 ?

Q12:

Given that π‘₯ = 2 is a solution of the equation π‘₯ + 𝑏 π‘₯ βˆ’ 2 4 = 0 2 , find the value of 𝑏 .

Q13:

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

  • A π‘₯ + 1 2 π‘₯ + 5 2 = 0 2
  • B π‘₯ βˆ’ 1 2 π‘₯ + 5 2 = 0 2
  • C π‘₯ βˆ’ 1 2 π‘₯ + 2 0 = 0 2
  • D π‘₯ βˆ’ 2 0 π‘₯ + 5 2 = 0 2
  • E π‘₯ βˆ’ 2 0 π‘₯ + 2 0 = 0 2

Q14:

Given that 3 is a solution of the equation 9 π‘₯ + 7 π‘₯ + π‘˜ = 0 2 , find the value of π‘˜ .

Q15:

Without solving the equation βˆ’ 3 π‘₯ βˆ’ 1 6 π‘₯ + 6 3 = 0 2 , find the sum of its roots.

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

Q16:

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

Q17:

Given that the sum of the roots of the equation 8 π‘₯ + 𝑏 π‘₯ + 1 8 = 0 2 is equal to their product, find the value of 𝑏 .

Q18:

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

  • A 6 π‘₯ βˆ’ 4 9 π‘₯ + 6 5 = 0 2
  • B 6 π‘₯ + 4 9 π‘₯ + 6 5 = 0 2
  • C 2 π‘₯ βˆ’ 3 π‘₯ βˆ’ 6 5 = 0 2
  • D 6 π‘₯ + 2 9 π‘₯ βˆ’ 6 5 = 0 2

Q19:

What is the simplest form of the quadratic equation whose roots are βˆ’ 6 and 1 4 5 ?

  • A 5 π‘₯ + 1 6 π‘₯ βˆ’ 8 4 = 0 2
  • B 5 π‘₯ βˆ’ 1 6 π‘₯ βˆ’ 8 4 = 0 2
  • C π‘₯ + 2 π‘₯ βˆ’ 2 4 = 0 2
  • D 5 π‘₯ βˆ’ 4 4 π‘₯ + 8 4 = 0 2

Q20:

Given that βˆ’ 1 is one of the roots of the equation π‘₯ + π‘Ž π‘₯ + 2 = 0 2 , find the value of π‘Ž and the value of the other root.

  • A π‘Ž = 3 , other root = βˆ’ 2
  • B π‘Ž = βˆ’ 3 , other root = 2
  • C π‘Ž = βˆ’ 3 , other root = βˆ’ 2
  • D π‘Ž = 3 , other root = 2

Q21:

What is the product of the roots of the quadratic equation π‘Ž π‘₯ + 𝑏 π‘₯ + 𝑐 = 0 2 ?

  • A 𝑐 π‘Ž
  • B βˆ’ 𝑏 𝑐
  • C π‘Ž 𝑏
  • D βˆ’ 𝑏 π‘Ž

Q22:

If the two roots of the equation βˆ’ 5 π‘₯ + 1 0 π‘₯ + π‘˜ = 0 2 are equal, determine the value of π‘˜ , then find the two roots.

  • A π‘˜ = βˆ’ 5 , roots: 1, 1
  • B π‘˜ = 5 , roots: 1, 1
  • C π‘˜ = 5 , roots: βˆ’ 1 , βˆ’ 1
  • D π‘˜ = βˆ’ 5 , roots: βˆ’ 1 , βˆ’ 1

Q23:

Given that a root of the equation π‘₯ + 1 8 π‘₯ + π‘˜ = 0 2 is π‘₯ = βˆ’ 3 , what is the value of π‘˜ ?

Q24:

Find, in its simplest form, the quadratic equation whose roots are π‘š + 3 𝑛 and π‘š βˆ’ 3 𝑛 .

  • A π‘₯ βˆ’ 2 π‘š π‘₯ + π‘š βˆ’ 9 𝑛 = 0 2 2 2
  • B π‘₯ βˆ’ 6 𝑛 π‘₯ + π‘š βˆ’ 9 𝑛 = 0 2 2 2
  • C π‘₯ βˆ’ 2 π‘š π‘₯ + π‘š βˆ’ 3 𝑛 = 0 2 2 2
  • D π‘₯ + 2 π‘₯ + π‘š βˆ’ 9 𝑛 = 0 2 2 2
  • E π‘₯ βˆ’ 2 π‘₯ + π‘š + 9 𝑛 = 0 2 2 2

Q25:

Find, in its simplest form, the quadratic equation whose roots are π‘š + 8 𝑛 and π‘š βˆ’ 8 𝑛 .

  • A π‘₯ βˆ’ 2 π‘š π‘₯ + π‘š βˆ’ 6 4 𝑛 = 0 2 2 2
  • B π‘₯ βˆ’ 1 6 𝑛 π‘₯ + π‘š βˆ’ 6 4 𝑛 = 0 2 2 2
  • C π‘₯ βˆ’ 2 π‘š π‘₯ + π‘š βˆ’ 8 𝑛 = 0 2 2 2
  • D π‘₯ + 2 π‘₯ + π‘š βˆ’ 6 4 𝑛 = 0 2 2 2
  • E π‘₯ βˆ’ 2 π‘₯ + π‘š + 6 4 𝑛 = 0 2 2 2
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