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Worksheet: Solving Quadratic Equation with Complex Roots

Q1:

Solve the equation 5 π‘₯ + 1 = βˆ’ 3 1 9 2 .

  • A π‘₯ = 8 √ 5 𝑖 , π‘₯ = βˆ’ 8 √ 5 𝑖
  • B π‘₯ = 8 , π‘₯ = βˆ’ 8
  • C π‘₯ = 8 √ 5 5 𝑖 , π‘₯ = βˆ’ 8 √ 5 5 𝑖
  • D π‘₯ = 8 𝑖 , π‘₯ = βˆ’ 8 𝑖
  • E π‘₯ = 8 5 𝑖 , π‘₯ = βˆ’ 8 5 𝑖

Q2:

Factor π‘₯ + 9 2 over the complex numbers.

  • A ( π‘₯ + 3 𝑖 ) ( π‘₯ + 3 𝑖 )
  • B ( π‘₯ + 3 ) ( π‘₯ βˆ’ 3 )
  • C ( π‘₯ βˆ’ 3 𝑖 ) ( π‘₯ βˆ’ 3 𝑖 )
  • D ( π‘₯ + 3 𝑖 ) ( π‘₯ βˆ’ 3 𝑖 )
  • E ( π‘₯ + 3 ) ( π‘₯ + 3 )

Q3:

Which quadratic equation has roots π‘₯ = Β± 3 𝑖 ?

  • A π‘₯ = βˆ’ 3 2
  • B π‘₯ = 9 2
  • C π‘₯ = 3 2
  • D π‘₯ = βˆ’ 9 2
  • E π‘₯ = βˆ’ 6 2

Q4:

Solve the equation π‘₯ = βˆ’ 1 2 .

  • A π‘₯ = βˆ’ 1 2
  • B π‘₯ = 1 , π‘₯ = βˆ’ 1
  • C π‘₯ = 𝑖 2 , π‘₯ = βˆ’ 𝑖 2
  • D π‘₯ = 𝑖 , π‘₯ = βˆ’ 𝑖
  • E π‘₯ = 𝑖 , π‘₯ = 1

Q5:

Which of the following best describes the roots of the equation π‘₯ + 1 7 = 0 2 ?

  • A two non-real roots
  • B one repeated real root
  • C one repeated non-real root
  • D two complex roots

Q6:

Solve the quadratic equation π‘₯ βˆ’ 4 π‘₯ + 8 = 0 2 .

  • A π‘₯ = 2 + 2 √ 3 , π‘₯ = 2 βˆ’ 2 √ 3
  • B π‘₯ = βˆ’ 2 + 2 𝑖 , π‘₯ = βˆ’ 2 βˆ’ 2 𝑖
  • C π‘₯ = βˆ’ 2 + 2 √ 3 , π‘₯ = βˆ’ 2 βˆ’ 2 √ 3
  • D π‘₯ = 2 + 2 𝑖 , π‘₯ = 2 βˆ’ 2 𝑖
  • E π‘₯ = 4 + 4 𝑖 , π‘₯ = 4 βˆ’ 4 𝑖

Q7:

Find the value of 𝑐 such that the quadratic equation 4 π‘₯ + 1 2 π‘₯ + 𝑐 = 0 2 has the roots βˆ’ 3 2 Β± 𝑖 .

Q8:

By completing the square, solve the equation π‘₯ + π‘₯ + 1 = 0 2 .

  • A π‘₯ = βˆ’ 1 2 + √ 3 2 𝑖
  • B π‘₯ = 1 2 , π‘₯ = βˆ’ 1 2
  • C π‘₯ = 1 , π‘₯ = βˆ’ 1
  • D π‘₯ = βˆ’ 1 2 + √ 3 2 𝑖 , π‘₯ = βˆ’ 1 2 βˆ’ √ 3 2 𝑖
  • E π‘₯ = βˆ’ √ 3 2 + 1 2 𝑖 , π‘₯ = βˆ’ √ 3 2 βˆ’ 1 2 𝑖

Q9:

If the discriminant of a quadratic equation is negative, will its complex roots be a conjugate pair?

  • Ayes
  • Bno

Q10:

Which of the following best describes the roots of the equation π‘₯ βˆ’ 1 7 = 0 2 ?

  • A two non-real roots
  • B one repeated real root
  • C one repeated non-real root
  • D two real roots

Q11:

The complex numbers π‘Ž + 𝑏 𝑖 and 𝑐 + 𝑑 𝑖 , where π‘Ž , 𝑏 , 𝑐 , and 𝑑 are real numbers, are roots of a quadratic polynomial with real coefficients. Given that 𝑏 β‰  0 , what conditions, if any, must π‘Ž , 𝑏 , 𝑐 , and 𝑑 satisfy?

  • A π‘Ž = 0 and 𝑐 = 0
  • B 𝑐 = π‘Ž and 𝑑 = 𝑏
  • Cthere are no additional conditions
  • D 𝑐 = π‘Ž and 𝑑 = βˆ’ 𝑏

Q12:

Factor π‘₯ + 𝑦 2 2 over the complex numbers.

  • A ( π‘₯ + 𝑦 𝑖 ) ( π‘₯ + 𝑦 𝑖 )
  • B ( π‘₯ + 𝑦 ) ( π‘₯ βˆ’ 𝑦 )
  • C ( π‘₯ βˆ’ 𝑦 𝑖 ) ( π‘₯ βˆ’ 𝑦 𝑖 )
  • D ( π‘₯ + 𝑦 𝑖 ) ( π‘₯ βˆ’ 𝑦 𝑖 )
  • E ( π‘₯ + 𝑦 ) ( π‘₯ + 𝑦 )