Worksheet: Solving Quadratic Equation 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−8𝑥+185=0 over the set of complex numbers.

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

Q2:

Which of the following best describes the roots of the equation 𝑥+17=0?

  • Atwo complex roots
  • Btwo non-real roots
  • Cone repeated real root
  • Done repeated non-real root

Q3:

Solve the equation 5𝑥+1=−319.

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

Q4:

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

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

Q5:

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

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

Q6:

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

  • A 2 , − 2
  • B 2 𝑖 , − 2 𝑖
  • C 2 𝑖
  • D4
  • E 4 𝑖 , − 4 𝑖

Q7:

Find the solution set of 𝑥+5=0.

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

Q8:

Find the solution set of 𝑥+8𝑥+185=0 given 𝑥∈ℂ.

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

Q9:

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

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

Q10:

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

  • AYes
  • BNo

Q11:

Factor 𝑥+9 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 )

Q12:

Solve the equation 𝑥=−1.

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

Q13:

Which quadratic equation has roots 𝑥=±3𝑖?

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

Q14:

Solve the quadratic equation 𝑥+𝑥+1=0.

  • A 𝑥 = − 1 + √ 3 𝑖 2 , 𝑥 = − 1 − √ 3 𝑖 2
  • B 𝑥 = − 1 + √ 3 𝑖 , 𝑥 = − 1 − √ 3 𝑖
  • C 𝑥 = 1 + √ 3 𝑖 2 , 𝑥 = 1 − √ 3 𝑖 2
  • D 𝑥 = − 1 + √ 5 2 , 𝑥 = − 1 − √ 5 2
  • E 𝑥 = 1 + √ 5 2 , 𝑥 = 1 − √ 5 2

Q15:

Solve the quadratic equation 4𝑥+3𝑥+1=0.

  • A 𝑥 = − 3 + √ 5 2 , 𝑥 = − 3 − √ 5 2
  • B 𝑥 = − 3 + √ 7 𝑖 4 , 𝑥 = − 3 − √ 7 𝑖 4
  • C 𝑥 = − 3 + √ 7 𝑖 8 , 𝑥 = − 3 − √ 7 𝑖 8
  • D 𝑥 = 1 , 𝑥 = − 1 4
  • E 𝑥 = 3 + √ 7 𝑖 8 , 𝑥 = 3 − √ 7 𝑖 8

Q16:

Solve the quadratic equation 𝑥−4𝑥+8=0.

  • A 𝑥 = 4 + 4 𝑖 , 𝑥 = 4 − 4 𝑖
  • B 𝑥 = 2 + 2 𝑖 , 𝑥 = 2 − 2 𝑖
  • C 𝑥 = 2 + 2 √ 3 , 𝑥 = 2 − 2 √ 3
  • D 𝑥 = − 2 + 2 𝑖 , 𝑥 = − 2 − 2 𝑖
  • E 𝑥 = − 2 + 2 √ 3 , 𝑥 = − 2 − 2 √ 3

Q17:

Find the value of 𝑐 such that the quadratic equation 4𝑥+12𝑥+𝑐=0 has the roots −32±𝑖.

Q18:

By completing the square, solve the equation 𝑥+𝑥+1=0.

  • A 𝑥 = − 1 2 + √ 3 2 𝑖
  • B 𝑥 = − √ 3 2 + 1 2 𝑖 , 𝑥 = − √ 3 2 − 1 2 𝑖
  • C 𝑥 = − 1 2 + √ 3 2 𝑖 , 𝑥 = − 1 2 − √ 3 2 𝑖
  • D 𝑥 = 1 , 𝑥 = − 1
  • E 𝑥 = 1 2 , 𝑥 = − 1 2

Q19:

The product of the roots of the equation 3𝑥+8𝑥+𝑘=0 is 4. Find the value of 𝑘 and the solution set of the equation.

  • A 𝑘 = 4 3 , − 4 3 + √ 3 4 3 𝑖 , − 4 3 − √ 3 4 3 𝑖
  • B 𝑘 = 2 4 , − 3 2 3 + √ 3 4 3 𝑖 , − 3 2 3 − √ 3 4 3 𝑖
  • C 𝑘 = 1 2 , − 4 3 + 2 √ 5 3 𝑖 , − 4 3 − 2 √ 5 3 𝑖
  • D 𝑘 = 4 , − 1 1 8 + √ 5 8 1 4 4 𝑖 , − 1 1 8 − √ 5 8 1 4 4 𝑖

Q20:

Factor 𝑥+42 over the complex numbers.

  • A ( 𝑥 + √ 4 2 ) ( 𝑥 + √ 4 2 )
  • B ( 𝑥 − √ 4 2 𝑖 ) ( 𝑥 − √ 4 2 𝑖 )
  • C ( 𝑥 + √ 4 2 𝑖 ) ( 𝑥 + √ 4 2 𝑖 )
  • D ( 𝑥 + √ 4 2 𝑖 ) ( 𝑥 − √ 4 2 𝑖 )
  • E ( 𝑥 + √ 4 2 ) ( 𝑥 − √ 4 2 )

Q21:

The complex numbers 𝑎+𝑏𝑖 and 𝑐+𝑑𝑖, where 𝑎, 𝑏, 𝑐, and 𝑑 are real numbers, are roots of a quadratic polynomial with real coefficients. Given that 𝑏≠0, what conditions, if any, must 𝑎, 𝑏, 𝑐, and 𝑑 satisfy?

  • Athere are no additional conditions
  • B 𝑐 = 𝑎 and 𝑑=−𝑏
  • C 𝑎 = 0 and 𝑐=0
  • D 𝑐 = 𝑎 and 𝑑=𝑏

Q22:

Determine the type of the roots of the equation (2𝑥−4)+17=0.

  • Areal and different
  • Breal and equal
  • Ccomplex and not real

Q23:

Given that 𝑓(𝑥)=𝑎𝑥+𝑏𝑥+𝑐 has a zero at 3−4𝑖 and 𝑓(0)=100, determine the values of 𝑎, 𝑏, and 𝑐.

  • A 𝑎 = − 1 4 . 2 9 , 𝑏 = − 8 5 . 7 4 , 𝑐 = − 1 0 0
  • B 𝑎 = 4 , 𝑏 = 2 4 , 𝑐 = 8 0
  • C 𝑎 = 4 , 𝑏 = − 2 4 , 𝑐 = 1 0 0
  • D 𝑎 = − 1 4 . 2 9 , 𝑏 = 8 5 . 7 4 , 𝑐 = 1 0 0
  • E 𝑎 = 3 , 𝑏 = − 4 , 𝑐 = 1 0 0

Q24:

Given that 𝑥=−4+𝑖 is one of the roots of the equation 6𝑥+48𝑥+𝑘=0, find the other root and the value of 𝑘.

  • A 𝑥 = − 4 − 𝑖 , 𝑘 = 1 5
  • B 𝑥 = 5 2 , 𝑘 = 1 6
  • C 𝑥 = − 4 − 𝑖 , 𝑘 = 1 6
  • D 𝑥 = − 4 − 𝑖 , 𝑘 = 1 0 2
  • E 𝑥 = 5 2 , 𝑘 = 1 5

Q25:

Find all possible values of 𝑧, where 𝑧∈ℂ, for which 8𝑧=𝑧+12.

  • A6, 2
  • B6, 2, 2√15𝑖+4, −2√15𝑖+4
  • C 2 √ 1 5 𝑖 + 4 , − 2 √ 1 5 𝑖 + 4
  • D6, 2, 2√15𝑖−4, −2√15𝑖−4
  • E 2 √ 1 5 𝑖 − 4 , − 2 √ 1 5 𝑖 − 4

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