Worksheet: Solving Quadratic Equations with Complex Coefficients

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 .

  • 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 , 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 , 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 .

  • 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 .

  • 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 .

  • 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 .

  • 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 .

  • 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 .

  • 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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