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{−4−6𝑖,−4+6𝑖}
  • B{−2−3𝑖,−2+3𝑖}
  • C{4−6𝑖,4+6𝑖}
  • D{4+13𝑖,4−13𝑖}

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.

  • A𝑥=8√55𝑖,𝑥=−8√55𝑖
  • B𝑥=8,𝑥=−8
  • C𝑥=85𝑖,𝑥=−85𝑖
  • D𝑥=8𝑖,𝑥=−8𝑖
  • E𝑥=8√5𝑖,𝑥=−8√5𝑖

Q4:

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

  • A−45+325𝑖
  • B−21225+14125𝑖
  • C21225−14125𝑖
  • D45−325𝑖

Q5:

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

  • A2+2√5𝑖,2−2√5𝑖
  • B−4+2√2𝑖,−4−2√2𝑖
  • C−2+√2𝑖,−2−√2𝑖
  • D4+2√2𝑖,4−2√2𝑖
  • E−2+2√5𝑖,−2−2√5𝑖

Q6:

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

  • A2,−2
  • B2𝑖,−2𝑖
  • C2𝑖
  • D4
  • E4𝑖,−4𝑖

Q7:

Find the solution set of 𝑥+5=0.

  • A√5,−√5
  • B52𝑖,−52𝑖
  • C√5𝑖,−√5𝑖
  • D52,−52

Q8:

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

  • A{4−6𝑖,4+6𝑖}
  • B{−4−6𝑖,−4+6𝑖}
  • C{2−3𝑖,2+3𝑖}
  • D{−4+13𝑖,−4−13𝑖}

Q9:

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

  • A512−√9512𝑖,512+√9512𝑖
  • B−124+√37120𝑖,−124−√37120𝑖
  • C512−√11512𝑖,512+√11512𝑖
  • D2512−√11512𝑖,2512+√11512𝑖

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

Q12:

Solve the equation 𝑥=−1.

  • A𝑥=1, 𝑥=−1
  • B𝑥=𝑖2, 𝑥=−𝑖2
  • C𝑥=𝑖, 𝑥=−𝑖
  • D𝑥=𝑖, 𝑥=1
  • E𝑥=−12

Q13:

Which quadratic equation has roots 𝑥=±3𝑖?

  • A𝑥=9
  • B𝑥=3
  • C𝑥=−3
  • D𝑥=−6
  • E𝑥=−9

Q14:

Solve the quadratic equation 𝑥+𝑥+1=0.

  • A𝑥=−1+√3𝑖2,𝑥=−1−√3𝑖2
  • B𝑥=−1+√3𝑖,𝑥=−1−√3𝑖
  • C𝑥=1+√3𝑖2,𝑥=1−√3𝑖2
  • D𝑥=−1+√52,𝑥=−1−√52
  • E𝑥=1+√52,𝑥=1−√52

Q15:

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

  • A𝑥=−3+√52,𝑥=−3−√52
  • B𝑥=−3+√7𝑖4,𝑥=−3−√7𝑖4
  • C𝑥=−3+√7𝑖8,𝑥=−3−√7𝑖8
  • D𝑥=1,𝑥=−14
  • E𝑥=3+√7𝑖8,𝑥=3−√7𝑖8

Q16:

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

  • A𝑥=4+4𝑖,𝑥=4−4𝑖
  • B𝑥=2+2𝑖,𝑥=2−2𝑖
  • C𝑥=2+2√3,𝑥=2−2√3
  • D𝑥=−2+2𝑖,𝑥=−2−2𝑖
  • E𝑥=−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.

  • A𝑥=−12+√32𝑖
  • B𝑥=−√32+12𝑖, 𝑥=−√32−12𝑖
  • C𝑥=−12+√32𝑖, 𝑥=−12−√32𝑖
  • D𝑥=1, 𝑥=−1
  • E𝑥=12, 𝑥=−12

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.

  • A𝑘=43, −43+√343𝑖,−43−√343𝑖
  • B𝑘=24, −323+√343𝑖,−323−√343𝑖
  • C𝑘=12, −43+2√53𝑖,−43−2√53𝑖
  • D𝑘=4, −118+√58144𝑖,−118−√58144𝑖

Q20:

Factor 𝑥+42 over the complex numbers.

  • A(𝑥+√42)(𝑥+√42)
  • B(𝑥−√42𝑖)(𝑥−√42𝑖)
  • C(𝑥+√42𝑖)(𝑥+√42𝑖)
  • D(𝑥+√42𝑖)(𝑥−√42𝑖)
  • E(𝑥+√42)(𝑥−√42)

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
  • B𝑐=𝑎 and 𝑑=−𝑏
  • C𝑎=0 and 𝑐=0
  • D𝑐=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.

  • A𝑎=−14.29, 𝑏=−85.74, 𝑐=−100
  • B𝑎=4, 𝑏=24, 𝑐=80
  • C𝑎=4, 𝑏=−24, 𝑐=100
  • D𝑎=−14.29, 𝑏=85.74, 𝑐=100
  • E𝑎=3, 𝑏=−4, 𝑐=100

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.

  • A𝑥=−4−𝑖, 𝑘=15
  • B𝑥=52, 𝑘=16
  • C𝑥=−4−𝑖, 𝑘=16
  • D𝑥=−4−𝑖, 𝑘=102
  • E𝑥=52, 𝑘=15

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
  • C2√15𝑖+4, −2√15𝑖+4
  • D6, 2, 2√15𝑖−4, −2√15𝑖−4
  • E2√15𝑖−4, −2√15𝑖−4

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