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Worksheet: Quadratic Equations with Complex Coefficients

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

Q1:

Given that ( π‘₯ + 6 𝑖 𝑦 ) + 4 0 ( 3 βˆ’ 𝑖 ) 3 + 𝑖 = 0 2 , where π‘₯ and 𝑦 are real numbers, find all the possible values of π‘₯ and 𝑦 .

  • A π‘₯ = 3 , 𝑦 = 1
  • B π‘₯ = 4 , 𝑦 = 1 or π‘₯ = βˆ’ 4 , 𝑦 = βˆ’ 1
  • C π‘₯ = 1 , 𝑦 = 0
  • D π‘₯ = 2 , 𝑦 = 1 or π‘₯ = βˆ’ 2 , 𝑦 = βˆ’ 1

Q2:

Determine all the real values of π‘₯ and 𝑦 that satisfy the equation 8 6 𝑖 = ( π‘₯ βˆ’ 2 2 𝑖 ) ( 𝑦 βˆ’ 𝑖 ) βˆ’ 3 8 .

  • A π‘₯ = 1 0 1 1 , 𝑦 = βˆ’ 6 6 5 or π‘₯ = 3 , 𝑦 = βˆ’ 4
  • B π‘₯ = 1 0 1 1 , 𝑦 = 6 6 5 or π‘₯ = 3 , 𝑦 = 4
  • C π‘₯ = βˆ’ 6 6 , 𝑦 = βˆ’ 6 6 5 or π‘₯ = βˆ’ 2 0 , 𝑦 = βˆ’ 4
  • D π‘₯ = βˆ’ 6 6 , 𝑦 = βˆ’ 1 0 1 1 or π‘₯ = βˆ’ 2 0 , 𝑦 = βˆ’ 3

Q3:

Given that π‘₯ = βˆ’ 4 + 𝑖 is one of the roots of the equation 6 π‘₯ + 4 8 π‘₯ + π‘˜ = 0 2 , find the other root and the value of π‘˜ .

  • A π‘₯ = 5 2 , π‘˜ = 1 6
  • B π‘₯ = βˆ’ 4 βˆ’ 𝑖 , π‘˜ = 1 5
  • C π‘₯ = 5 2 , π‘˜ = 1 5
  • D π‘₯ = βˆ’ 4 βˆ’ 𝑖 , π‘˜ = 1 0 2
  • E π‘₯ = βˆ’ 4 βˆ’ 𝑖 , π‘˜ = 1 6

Q4:

Find the solution set of the equation in .

  • A
  • B
  • C
  • D
  • E

Q5:

Find the solution set of the equation ( 1 βˆ’ 𝑖 ) π‘₯ βˆ’ ( 8 βˆ’ 4 𝑖 ) π‘₯ + 5 + 7 𝑖 = 0 2 in β„‚ .

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

Q6:

Find the solution set of the equation ( 1 + 𝑖 ) π‘₯ βˆ’ ( 6 + 2 𝑖 ) π‘₯ + 3 βˆ’ 5 𝑖 = 0 2 in β„‚ .

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

Q7:

Find the solution set of the equation ( 1 βˆ’ 𝑖 ) π‘₯ βˆ’ ( 6 βˆ’ 2 𝑖 ) π‘₯ + 7 + 𝑖 = 0 2 in β„‚ .

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

Q8:

Find the solution set of the equation ( 1 βˆ’ 𝑖 ) π‘₯ βˆ’ ( 6 βˆ’ 2 𝑖 ) π‘₯ + 6 + 2 𝑖 = 0 2 in β„‚ .

  • 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 }

Q9:

Find the solution set of the equation ( 1 βˆ’ 𝑖 ) π‘₯ βˆ’ ( 6 βˆ’ 2 𝑖 ) π‘₯ + 3 + 5 𝑖 = 0 2 in β„‚ .

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

Q10:

Find the solution set of the equation ( 1 + 𝑖 ) π‘₯ βˆ’ ( 6 + 2 𝑖 ) π‘₯ + 7 βˆ’ 𝑖 = 0 2 in β„‚ .

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

Q11:

Find the solution set of the equation ( 1 + 𝑖 ) π‘₯ βˆ’ ( 6 + 2 𝑖 ) π‘₯ + 6 βˆ’ 2 𝑖 = 0 2 in β„‚ .

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

Q12:

Find the solution set of the equation ( 1 + 𝑖 ) π‘₯ βˆ’ ( 8 + 4 𝑖 ) π‘₯ + 5 βˆ’ 7 𝑖 = 0 2 in β„‚ .

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

Q13:

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

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

Q14:

Find the solution set of ( π‘₯ + 6 ) βˆ’ 2 ( π‘₯ + 6 ) + 1 = 0 6 3 in β„‚ .

  • A { βˆ’ 5 }
  • B  7 , πœ” βˆ’ 6 , πœ” βˆ’ 6  2
  • C { βˆ’ 5 , 5 }
  • D  βˆ’ 5 , πœ” βˆ’ 6 , πœ” βˆ’ 6  2

Q15:

Given that ( π‘₯ + 𝑦 𝑖 ) = 2 βˆ’ 2 𝑖 βˆ’ 1 βˆ’ 𝑖 2 , find all possible real values of π‘₯ and 𝑦 .

  • A { ( βˆ’ 3 , 1 ) , ( 3 , βˆ’ 1 ) }
  • B { ( 0 , 2 ) , ( 0 , βˆ’ 2 ) }
  • C  ο€Ό βˆ’ 1 2 , 1 2  , ο€Ό 1 2 , βˆ’ 1 2  
  • D { ( 1 , 1 ) , ( βˆ’ 1 , βˆ’ 1 ) }
  • E  ο€» √ 2 , √ 2  , ο€» βˆ’ √ 2 , βˆ’ √ 2  

Q16:

Find the solution set of π‘₯ + 4 πœ” π‘₯ βˆ’ 4 πœ” = 0 2 in β„‚ .

  • A { 2 πœ” , βˆ’ 2 πœ” }
  • B { 2 ( πœ” βˆ’ 𝑖 ) , 2 ( πœ” + 𝑖 ) }
  • C { 2 πœ” 𝑖 , βˆ’ 2 πœ” 𝑖 }
  • D { βˆ’ 2 ( πœ” βˆ’ 𝑖 ) , βˆ’ 2 ( πœ” + 𝑖 ) }

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