Solve polynomial equations with complex roots

A question is this type if and only if it asks to solve a polynomial equation (typically quadratic or cubic) with complex coefficients or to find complex roots, giving answers in Cartesian form x + iy.

7 questions · Standard +0.4

Sort by: Default | Easiest first | Hardest first
CAIE P3 2023 March Q4
5 marks Standard +0.8
4 Solve the equation $$\frac { 5 z } { 1 + 2 \mathrm { i } } - z z ^ { * } + 30 + 10 \mathrm { i } = 0$$ giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
CAIE P3 2022 November Q6
6 marks Standard +0.8
6 Solve the quadratic equation \(( 1 - 3 \mathrm { i } ) z ^ { 2 } - ( 2 + \mathrm { i } ) z + \mathrm { i } = 0\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
OCR FP3 2007 January Q3
7 marks Standard +0.3
3
  1. Solve the equation \(z ^ { 2 } - 6 z + 36 = 0\), and give your answers in the form \(r ( \cos \theta \pm \mathrm { i } \sin \theta )\), where \(r > 0\) and \(0 \leqslant \theta \leqslant \pi\).
  2. Given that \(Z\) is either of the roots found in part (i), deduce the exact value of \(Z ^ { - 3 }\).
OCR FP3 2010 January Q4
7 marks Standard +0.8
4
  1. Write down, in cartesian form, the roots of the equation \(z ^ { 4 } = 16\).
  2. Hence solve the equation \(w ^ { 4 } = 16 ( 1 - w ) ^ { 4 }\), giving your answers in cartesian form.
OCR Further Pure Core 2 2024 June Q2
6 marks Moderate -0.3
2 In this question you must show detailed reasoning.
  1. Solve the equation \(x ^ { 2 } - 6 x + 58 = 0\). Give your solutions in the form \(a + b\) i where \(a\) and \(b\) are real numbers.
  2. Determine, in exact form, \(\arg ( - 10 + ( 5 \sqrt { 12 } ) \mathrm { i } ) ^ { 5 }\).
AQA FP3 Q6
17 marks Challenging +1.2
6 It is given that \(z = \mathrm { e } ^ { \mathrm { i } \theta }\).
    1. Show that $$z + \frac { 1 } { z } = 2 \cos \theta$$
    2. Find a similar expression for $$z ^ { 2 } + \frac { 1 } { z ^ { 2 } }$$ (2 marks)
    3. Hence show that $$z ^ { 2 } - z + 2 - \frac { 1 } { z } + \frac { 1 } { z ^ { 2 } } = 4 \cos ^ { 2 } \theta - 2 \cos \theta$$ (3 marks)
  2. Hence solve the quartic equation $$z ^ { 4 } - z ^ { 3 } + 2 z ^ { 2 } - z + 1 = 0$$ giving the roots in the form \(a + \mathrm { i } b\).
AQA Further Paper 1 2019 June Q4
4 marks Moderate -0.5
4 Solve the equation \(2 z - 5 \mathrm { i } z ^ { * } = 12\)