Worksheet: Solving Quadratic Equations with Complex Coefficients

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In this worksheet, we will practice solving quadratic equations with complex coefficients using the quadratic formula.

Q1:

Find the solution set of the equation ( 1 − 𝑖 ) 𝑥 − ( 8 − 4 𝑖 ) 𝑥 + 5 + 7 𝑖 = 0 2 in ℂ .

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

Q2:

Given that ( 𝑥 + 6 𝑖 𝑦 ) + 4 0 ( 3 − 𝑖 ) 3 + 𝑖 = 0 2 , where 𝑥 and 𝑦 are real numbers, find all the possible values of 𝑥 and 𝑦 .

  • A 𝑥 = 3 , 𝑦 = 1
  • B 𝑥 = 4 , 𝑦 = 1 or 𝑥 = − 4 , 𝑦 = − 1
  • C 𝑥 = 1 , 𝑦 = 0
  • D 𝑥 = 2 , 𝑦 = 1 or 𝑥 = − 2 , 𝑦 = − 1

Q3:

Given that 𝑥 = − 4 + 𝑖 is one of the roots of the equation 6 𝑥 + 4 8 𝑥 + 𝑘 = 0 2 , find the other root and the value of 𝑘 .

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

Q4:

Find the solution set of ( 𝑥 + 6 ) − 2 ( 𝑥 + 6 ) + 1 = 0 in terms of 𝜔 , where 𝜔 is a complex cube root of unity.

  • A { − 5 }
  • B 7 , 𝜔 − 6 , 𝜔 − 6
  • C { − 5 , 5 }
  • D − 5 , 𝜔 − 6 , 𝜔 − 6

Q5:

Find the solution set of 𝑥 + 4 𝜔 𝑥 − 4 𝜔 = 0 in ℂ , where 𝜔 is a complex cube root of unity.

  • A { 2 𝜔 , − 2 𝜔 }
  • B { 2 ( 𝜔 − 𝑖 ) , 2 ( 𝜔 + 𝑖 ) }
  • C { 2 𝜔 𝑖 , − 2 𝜔 𝑖 }
  • D { − 2 ( 𝜔 − 𝑖 ) , − 2 ( 𝜔 + 𝑖 ) }

Q6:

Determine all the real values of 𝑥 and 𝑦 that satisfy the equation 8 6 𝑖 = ( 𝑥 − 2 2 𝑖 ) ( 𝑦 − 𝑖 ) − 3 8 .

  • A 𝑥 = 1 0 1 1 , 𝑦 = − 6 6 5 or 𝑥 = 3 , 𝑦 = − 4
  • B 𝑥 = 1 0 1 1 , 𝑦 = 6 6 5 or 𝑥 = 3 , 𝑦 = 4
  • C 𝑥 = − 6 6 , 𝑦 = − 6 6 5 or 𝑥 = − 2 0 , 𝑦 = − 4
  • D 𝑥 = − 6 6 , 𝑦 = − 1 0 1 1 or 𝑥 = − 2 0 , 𝑦 = − 3

Q7:

Find the solution set of the equation in .

  • A
  • B
  • C
  • D
  • E

Q8:

Find all possible values of 𝑧 , where 𝑧 ∈ ℂ , for which 8 𝑧 = 𝑧 + 1 2 2 .

  • A 2 √ 1 5 𝑖 − 4 , − 2 √ 1 5 𝑖 − 4
  • B6, 2
  • C6, 2, 2 √ 1 5 𝑖 + 4 , − 2 √ 1 5 𝑖 + 4
  • D6, 2, 2 √ 1 5 𝑖 − 4 , − 2 √ 1 5 𝑖 − 4
  • E 2 √ 1 5 𝑖 + 4 , − 2 √ 1 5 𝑖 + 4

Q9:

Given that ( 𝑥 + 𝑦 𝑖 ) = 2 − 2 𝑖 − 1 − 𝑖 2 , find all possible real values of 𝑥 and 𝑦 .

  • A { ( − 3 , 1 ) , ( 3 , − 1 ) }
  • B { ( 0 , 2 ) , ( 0 , − 2 ) }
  • C − 1 2 , 1 2 , 1 2 , − 1 2
  • D { ( 1 , 1 ) , ( − 1 , − 1 ) }
  • E √ 2 , √ 2 , − √ 2 , − √ 2

Q10:

Determine the solution set of 1 4 𝑦 + 3 6 = 0 2 over the set of complex numbers.

  • A { − 1 2 𝑖 }
  • B { 1 2 𝑖 }
  • C ∅
  • D { − 1 2 𝑖 , 1 2 𝑖 }

Q11:

Find the solution set of the equation ( 1 + 𝑖 ) 𝑥 − ( 6 + 2 𝑖 ) 𝑥 + 3 − 5 𝑖 = 0 2 in ℂ .

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

Q12:

Find the solution set of the equation ( 1 − 𝑖 ) 𝑥 − ( 6 − 2 𝑖 ) 𝑥 + 7 + 𝑖 = 0 2 in ℂ .

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

Q13:

Find the solution set of the equation ( 1 − 𝑖 ) 𝑥 − ( 6 − 2 𝑖 ) 𝑥 + 6 + 2 𝑖 = 0 2 in ℂ .

  • A { 1 + 3 𝑖 , 1 + 𝑖 }
  • B { 2 + 2 𝑖 , 2 }
  • C { 2 + 2 𝑖 , 2 𝑖 }
  • D { 3 + 𝑖 , 1 + 𝑖 }
  • E { 3 + 2 𝑖 , 1 }

Q14:

Find the solution set of the equation ( 1 − 𝑖 ) 𝑥 − ( 6 − 2 𝑖 ) 𝑥 + 3 + 5 𝑖 = 0 2 in ℂ .

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

Q15:

Find the solution set of the equation ( 1 + 𝑖 ) 𝑥 − ( 6 + 2 𝑖 ) 𝑥 + 7 − 𝑖 = 0 2 in ℂ .

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

Q16:

Find the solution set of the equation ( 1 + 𝑖 ) 𝑥 − ( 6 + 2 𝑖 ) 𝑥 + 6 − 2 𝑖 = 0 2 in ℂ .

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

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