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Lesson: Solving Quadratic Equation with Complex Roots

In this lesson, we will learn how to solve quadratic equations whose roots are complex numbers.

Sample Question Videos

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01:17
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Worksheet • 17 Questions • 3 Videos

Q1:

Determine the solution set of π‘₯ βˆ’ 8 π‘₯ + 1 8 5 = 0 2 over the set of complex numbers.

  • A { 4 + 1 3 𝑖 , 4 βˆ’ 1 3 𝑖 }
  • B { 4 βˆ’ 6 𝑖 , 4 + 6 𝑖 }
  • C { βˆ’ 4 βˆ’ 6 𝑖 , βˆ’ 4 + 6 𝑖 }
  • D { βˆ’ 2 βˆ’ 3 𝑖 , βˆ’ 2 + 3 𝑖 }

Q2:

Which of the following best describes the roots of the equation π‘₯ + 1 7 = 0 2 ?

  • A two complex roots
  • B one repeated real root
  • C one repeated non-real root
  • D two non-real roots

Q3:

Factor π‘₯ + 𝑦 2 2 over the complex numbers.

  • A ( π‘₯ + 𝑦 𝑖 ) ( π‘₯ βˆ’ 𝑦 𝑖 )
  • B ( π‘₯ + 𝑦 ) ( π‘₯ + 𝑦 )
  • C ( π‘₯ + 𝑦 ) ( π‘₯ βˆ’ 𝑦 )
  • D ( π‘₯ βˆ’ 𝑦 𝑖 ) ( π‘₯ βˆ’ 𝑦 𝑖 )
  • E ( π‘₯ + 𝑦 𝑖 ) ( π‘₯ + 𝑦 𝑖 )

Q4:

Solve the equation 5 π‘₯ + 1 = βˆ’ 3 1 9 2 .

  • A π‘₯ = 8 𝑖 , π‘₯ = βˆ’ 8 𝑖
  • B π‘₯ = 8 5 𝑖 , π‘₯ = βˆ’ 8 5 𝑖
  • C π‘₯ = 8 , π‘₯ = βˆ’ 8
  • D π‘₯ = 8 √ 5 5 𝑖 , π‘₯ = βˆ’ 8 √ 5 5 𝑖
  • E π‘₯ = 8 √ 5 𝑖 , π‘₯ = βˆ’ 8 √ 5 𝑖

Q5:

Given that ( 8 βˆ’ 6 𝑖 ) is one of the roots of π‘₯ + 𝑏 π‘₯ + 6 = 0 2 , determine the value of 𝑏 .

  • A βˆ’ 2 1 2 2 5 + 1 4 1 2 5 𝑖
  • B βˆ’ 4 5 + 3 2 5 𝑖
  • C 4 5 βˆ’ 3 2 5 𝑖
  • D 2 1 2 2 5 βˆ’ 1 4 1 2 5 𝑖

Q6:

Find the roots of the quadratic equation ( π‘₯ + 4 ) + 8 = 0 2 .

  • A βˆ’ 4 + 2 √ 2 𝑖 , βˆ’ 4 βˆ’ 2 √ 2 𝑖
  • B βˆ’ 2 + √ 2 𝑖 , βˆ’ 2 βˆ’ √ 2 𝑖
  • C 4 + 2 √ 2 𝑖 , 4 βˆ’ 2 √ 2 𝑖
  • D 2 + 2 √ 5 𝑖 , 2 βˆ’ 2 √ 5 𝑖
  • E βˆ’ 2 + 2 √ 5 𝑖 , βˆ’ 2 βˆ’ 2 √ 5 𝑖

Q7:

Solve the equation 2 π‘₯ + 8 = 0 2 over the set of complex numbers.

  • A 2 𝑖 , βˆ’ 2 𝑖
  • B 4 𝑖 , βˆ’ 4 𝑖
  • C 2 𝑖
  • D4
  • E 2 , βˆ’ 2

Q8:

Find the solution set of π‘₯ + 5 = 0 2 .

  • A  √ 5 𝑖 , βˆ’ √ 5 𝑖 
  • B  √ 5 , βˆ’ √ 5 
  • C  5 2 , βˆ’ 5 2 
  • D  5 2 𝑖 , βˆ’ 5 2 𝑖 

Q9:

Find the solution set of π‘₯ + 8 π‘₯ + 1 8 5 = 0 2 given π‘₯ ∈ β„‚ .

  • A { βˆ’ 4 + 1 3 𝑖 , βˆ’ 4 βˆ’ 1 3 𝑖 }
  • B { βˆ’ 4 βˆ’ 6 𝑖 , βˆ’ 4 + 6 𝑖 }
  • C { 4 βˆ’ 6 𝑖 , 4 + 6 𝑖 }
  • D { 2 βˆ’ 3 𝑖 , 2 + 3 𝑖 }

Q10:

Find the solution set of βˆ’ 6 π‘₯ + 5 π‘₯ βˆ’ 5 = 0 2 over β„‚ .

  • A  5 1 2 βˆ’ √ 9 5 1 2 𝑖 , 5 1 2 + √ 9 5 1 2 𝑖 
  • B  5 1 2 βˆ’ √ 1 1 5 1 2 𝑖 , 5 1 2 + √ 1 1 5 1 2 𝑖 
  • C  βˆ’ 1 2 4 + √ 3 7 1 2 0 𝑖 , βˆ’ 1 2 4 βˆ’ √ 3 7 1 2 0 𝑖 
  • D  2 5 1 2 βˆ’ √ 1 1 5 1 2 𝑖 , 2 5 1 2 + √ 1 1 5 1 2 𝑖 

Q11:

Find the solution set of βˆ’ π‘₯ + 1 6 = 0 4 in the set of complex numbers.

  • A { 2 , βˆ’ 2 , 2 𝑖 , βˆ’ 2 𝑖 }
  • B { 2 , βˆ’ 2 }
  • C { 2 , 2 𝑖 }
  • D { 2 , βˆ’ 2 , 4 𝑖 , βˆ’ 4 𝑖 }
  • E { 4 , βˆ’ 4 𝑖 }

Q12:

Find the solution set of the equation π‘₯ + 3 = 0 2 in ℝ .

  • A βˆ…
  • B { ( 3 , βˆ’ 3 ) }
  • C { βˆ’ 3 }
  • D  βˆ’ √ 3 , √ 3 
  • E  βˆ’ √ 3 

Q13:

If the discriminant of a quadratic equation with real coefficients is negative, will its roots be a complex conjugate pair?

  • ANo
  • BYes

Q14:

Which of the following best describes the roots of the equation π‘₯ βˆ’ 1 7 = 0 2 ?

  • A two real roots
  • B one repeated real root
  • C one repeated non-real root
  • D two non-real roots

Q15:

Factor π‘₯ + 9 2 over the complex numbers.

  • A ( π‘₯ + 3 𝑖 ) ( π‘₯ βˆ’ 3 𝑖 )
  • B ( π‘₯ + 3 ) ( π‘₯ + 3 )
  • C ( π‘₯ + 3 ) ( π‘₯ βˆ’ 3 )
  • D ( π‘₯ βˆ’ 3 𝑖 ) ( π‘₯ βˆ’ 3 𝑖 )
  • E ( π‘₯ + 3 𝑖 ) ( π‘₯ + 3 𝑖 )

Q16:

Solve the equation π‘₯ = βˆ’ 1 2 .

  • A π‘₯ = 𝑖 , π‘₯ = βˆ’ 𝑖
  • B π‘₯ = 𝑖 , π‘₯ = 1
  • C π‘₯ = 1 , π‘₯ = βˆ’ 1
  • D π‘₯ = 𝑖 2 , π‘₯ = βˆ’ 𝑖 2
  • E π‘₯ = βˆ’ 1 2

Q17:

Which quadratic equation has roots π‘₯ = Β± 3 𝑖 ?

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