Worksheet: Quadratic Equations with Complex Roots

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In this worksheet, we will practice solving quadratic equations whose roots are complex numbers.

Q1:

Determine the solution set of 𝑥 − 8 𝑥 + 1 8 5 = 0 over the set of complex numbers.

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

Q2:

Which of the following best describes the roots of the equation 𝑥 + 1 7 = 0 2 ?

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

Q3:

Factor 𝑥 + 𝑦 2 2 over the complex numbers.

  • A ( 𝑥 + 𝑦 𝑖 ) ( 𝑥 + 𝑦 𝑖 )
  • B ( 𝑥 + 𝑦 ) ( 𝑥 − 𝑦 )
  • C ( 𝑥 − 𝑦 𝑖 ) ( 𝑥 − 𝑦 𝑖 )
  • D ( 𝑥 + 𝑦 𝑖 ) ( 𝑥 − 𝑦 𝑖 )
  • E ( 𝑥 + 𝑦 ) ( 𝑥 + 𝑦 )

Q4:

Solve the equation 5 𝑥 + 1 = − 3 1 9 2 .

  • A 𝑥 = 8 √ 5 𝑖 , 𝑥 = − 8 √ 5 𝑖
  • B 𝑥 = 8 , 𝑥 = − 8
  • C 𝑥 = 8 √ 5 5 𝑖 , 𝑥 = − 8 √ 5 5 𝑖
  • D 𝑥 = 8 𝑖 , 𝑥 = − 8 𝑖
  • E 𝑥 = 8 5 𝑖 , 𝑥 = − 8 5 𝑖

Q5:

Given that ( 8 − 6 𝑖 ) is one of the roots of 𝑥 + 𝑏 𝑥 + 6 = 0 2 , determine the value of 𝑏 .

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

Q6:

Find the roots of the quadratic equation ( 𝑥 + 4 ) + 8 = 0 2 .

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

Q7:

Solve the equation 2 𝑥 + 8 = 0 2 over the set of complex numbers.

  • A 2 , − 2
  • B 2 𝑖
  • C4
  • D 2 𝑖 , − 2 𝑖
  • E 4 𝑖 , − 4 𝑖

Q8:

Find the solution set of 𝑥 + 5 = 0 .

  • A 5 2 𝑖 , − 5 2 𝑖
  • B √ 5 , − √ 5
  • C 5 2 , − 5 2
  • D √ 5 𝑖 , − √ 5 𝑖

Q9:

Find the solution set of 𝑥 + 8 𝑥 + 1 8 5 = 0 given 𝑥 ∈ ℂ .

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

Q10:

Find the solution set of − 6 𝑥 + 5 𝑥 − 5 = 0 over ℂ .

  • A 2 5 1 2 − √ 1 1 5 1 2 𝑖 , 2 5 1 2 + √ 1 1 5 1 2 𝑖
  • B 5 1 2 − √ 1 1 5 1 2 𝑖 , 5 1 2 + √ 1 1 5 1 2 𝑖
  • C − 1 2 4 + √ 3 7 1 2 0 𝑖 , − 1 2 4 − √ 3 7 1 2 0 𝑖
  • D 5 1 2 − √ 9 5 1 2 𝑖 , 5 1 2 + √ 9 5 1 2 𝑖

Q11:

Find the solution set of − 𝑥 + 1 6 = 0 in the set of complex numbers.

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

Q12:

Find the solution set of the equation 𝑥 + 3 = 0 in ℝ .

  • A − √ 3
  • B { − 3 }
  • C − √ 3 , √ 3
  • D ∅
  • E { ( 3 , − 3 ) }

Q13:

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

  • AYes
  • BNo

Q14:

Which of the following best describes the roots of the equation 𝑥 − 1 7 = 0 2 ?

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

Q15:

Factor 𝑥 + 9 2 over the complex numbers.

  • A ( 𝑥 + 3 𝑖 ) ( 𝑥 + 3 𝑖 )
  • B ( 𝑥 + 3 ) ( 𝑥 − 3 )
  • C ( 𝑥 − 3 𝑖 ) ( 𝑥 − 3 𝑖 )
  • D ( 𝑥 + 3 𝑖 ) ( 𝑥 − 3 𝑖 )
  • E ( 𝑥 + 3 ) ( 𝑥 + 3 )

Q16:

Solve the equation 𝑥 = − 1 2 .

  • A 𝑥 = − 1 2
  • B 𝑥 = 1 , 𝑥 = − 1
  • C 𝑥 = 𝑖 2 , 𝑥 = − 𝑖 2
  • D 𝑥 = 𝑖 , 𝑥 = − 𝑖
  • E 𝑥 = 𝑖 , 𝑥 = 1

Q17:

Which quadratic equation has roots 𝑥 = ± 3 𝑖 ?

  • A 𝑥 = − 3 2
  • B 𝑥 = 9 2
  • C 𝑥 = 3 2
  • D 𝑥 = − 9 2
  • E 𝑥 = − 6 2

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