Worksheet: Solving Quadratic Equation with Complex Roots

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

Q1:

Determine the solution set of 𝑥8𝑥+185=0 over the set of complex numbers.

  • A{46𝑖,4+6𝑖}
  • B{23𝑖,2+3𝑖}
  • C{46𝑖,4+6𝑖}
  • D{4+13𝑖,413𝑖}

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=855𝑖,𝑥=855𝑖
  • B𝑥=8,𝑥=8
  • C𝑥=85𝑖,𝑥=85𝑖
  • D𝑥=8𝑖,𝑥=8𝑖
  • E𝑥=85𝑖,𝑥=85𝑖

Q4:

Given that (86𝑖) is one of the roots of 𝑥+𝑏𝑥+6=0, determine the value of 𝑏.

  • A45+325𝑖
  • B21225+14125𝑖
  • C2122514125𝑖
  • D45325𝑖

Q5:

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

  • A2+25𝑖,225𝑖
  • B4+22𝑖,422𝑖
  • C2+2𝑖,22𝑖
  • D4+22𝑖,422𝑖
  • E2+25𝑖,225𝑖

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.

  • A5,5
  • B52𝑖,52𝑖
  • C5𝑖,5𝑖
  • D52,52

Q8:

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

  • A{46𝑖,4+6𝑖}
  • B{46𝑖,4+6𝑖}
  • C{23𝑖,2+3𝑖}
  • D{4+13𝑖,413𝑖}

Q9:

Find the solution set of 6𝑥+5𝑥5=0 over .

  • A5129512𝑖,512+9512𝑖
  • B124+37120𝑖,12437120𝑖
  • C51211512𝑖,512+11512𝑖
  • D251211512𝑖,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)(𝑥3𝑖)
  • B(𝑥+3𝑖)(𝑥+3𝑖)
  • C(𝑥+3)(𝑥3)
  • D(𝑥+3)(𝑥+3)
  • E(𝑥3𝑖)(𝑥3𝑖)

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=13𝑖2
  • B𝑥=1+3𝑖,𝑥=13𝑖
  • C𝑥=1+3𝑖2,𝑥=13𝑖2
  • D𝑥=1+52,𝑥=152
  • E𝑥=1+52,𝑥=152

Q15:

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

  • A𝑥=3+52,𝑥=352
  • B𝑥=3+7𝑖4,𝑥=37𝑖4
  • C𝑥=3+7𝑖8,𝑥=37𝑖8
  • D𝑥=1,𝑥=14
  • E𝑥=3+7𝑖8,𝑥=37𝑖8

Q16:

Solve the quadratic equation 𝑥4𝑥+8=0.

  • A𝑥=4+4𝑖,𝑥=44𝑖
  • B𝑥=2+2𝑖,𝑥=22𝑖
  • C𝑥=2+23,𝑥=223
  • D𝑥=2+2𝑖,𝑥=22𝑖
  • E𝑥=2+23,𝑥=223

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=3212𝑖
  • C𝑥=12+32𝑖, 𝑥=1232𝑖
  • 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,43343𝑖
  • B𝑘=24, 323+343𝑖,323343𝑖
  • C𝑘=12, 43+253𝑖,43253𝑖
  • D𝑘=4, 118+58144𝑖,11858144𝑖

Q20:

Factor 𝑥+42 over the complex numbers.

  • A(𝑥+42)(𝑥+42)
  • B(𝑥42𝑖)(𝑥42𝑖)
  • C(𝑥+42𝑖)(𝑥+42𝑖)
  • D(𝑥+42𝑖)(𝑥42𝑖)
  • E(𝑥+42)(𝑥42)

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 𝑏0, 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 (2𝑥4)+17=0.

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

Q23:

Given that 𝑓(𝑥)=𝑎𝑥+𝑏𝑥+𝑐 has a zero at 34𝑖 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=4𝑖, 𝑘=15
  • B𝑥=52, 𝑘=16
  • C𝑥=4𝑖, 𝑘=16
  • D𝑥=4𝑖, 𝑘=102
  • E𝑥=52, 𝑘=15

Q25:

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

  • A6, 2
  • B6, 2, 215𝑖+4, 215𝑖+4
  • C215𝑖+4, 215𝑖+4
  • D6, 2, 215𝑖4, 215𝑖4
  • E215𝑖4, 215𝑖4

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