Lesson Worksheet: Solving Quadratic Equations with Complex Roots Mathematics

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 distinct 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 where 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:

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)

Q11:

Solve the equation 𝑥=1.

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

Q12:

Which quadratic equation has roots 𝑥=±3𝑖?

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

Q13:

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

Q14:

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

Q15:

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

Q16:

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

Q17:

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

Q18:

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

Q19:

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)

Q20:

Given that 𝑓(𝑥)=𝑎𝑥+𝑏𝑥+𝑐 has a zero at 34𝑖 and 𝑓(0)=100, determine the values of 𝑎, 𝑏, and 𝑐 where they are real numbers.

  • 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

Q21:

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

Q22:

The sum of the roots of the equation 𝑥+𝑘𝑥+7=0 is 1. Find the value of 𝑘 and the solution set of the equation.

  • A𝑘=1, 12+62𝑖,1262𝑖
  • B𝑘=1, 12+62𝑖,1262𝑖
  • C𝑘=1, 12+332𝑖,12332𝑖
  • D𝑘=1, 12+332𝑖,12332𝑖

Q23:

Solve 5𝑥4𝑥+4=0, where 𝑥.

  • A{(5+6𝑖),(5+𝑖)}
  • B25+45𝑖,2545𝑖
  • C25+45𝑖,2545𝑖
  • D{(5𝑖),(56𝑖)}
  • E3232𝑖,32+32𝑖

Q24:

Determine the solution set of 14𝑦+36=0 over the set of complex numbers.

  • A{12𝑖,12𝑖}
  • B{12𝑖}
  • C{12𝑖}
  • D

Q25:

Find, in its simplest form, the quadratic equation whose roots are 422𝑖53𝑖 and 4+46𝑖45𝑖.

  • A𝑥+12𝑥+52=0
  • B𝑥12𝑥+52=0
  • C𝑥12𝑥+20=0
  • D𝑥20𝑥+52=0
  • E𝑥20𝑥+20=0

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