Quadratic equations involving z² and z*

Equations of the form z² + az* + b = 0 or similar, requiring substitution z = x + iy to form a system where one equation is quadratic, typically solved by combining with the linear constraint from the imaginary part.

6 questions · Standard +0.9

4.02i Quadratic equations: with complex roots
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CAIE P3 2022 March Q6
6 marks Challenging +1.2
6 Find the complex numbers \(w\) which satisfy the equation \(w ^ { 2 } + 2 \mathrm { i } w ^ { * } = 1\) and are such that \(\operatorname { Re } w \leqslant 0\). Give your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
OCR MEI Further Pure Core 2020 November Q6
4 marks Standard +0.8
6 The complex number \(z\) satisfies the equation \(z ^ { 2 } - 4 \mathrm { i } z ^ { * } + 11 = 0\).
Given that \(\operatorname { Re } ( z ) > 0\), find \(z\) in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real numbers. Section B (108 marks)
Answer all the questions.
AQA FP1 2014 June Q4
6 marks Standard +0.3
Find the complex number \(z\) such that $$5iz + 3z^* + 16 = 8i$$ Give your answer in the form \(a + bi\), where \(a\) and \(b\) are real. [6 marks]
AQA Further Paper 1 2021 June Q6
10 marks Challenging +1.8
  1. Show that the equation $$(2z - z^*)^* = z^2$$ has exactly four solutions and state these solutions. [7 marks]
    1. Plot the four solutions to the equation in part (a) on the Argand diagram below and join them together to form a quadrilateral with one line of symmetry. [2 marks] \includegraphics{figure_6b}
    2. Show that the area of this quadrilateral is \(\frac{\sqrt{15}}{2}\) square units. [1 mark]
SPS SPS FM Pure 2023 November Q1
4 marks Standard +0.8
The complex number \(z\) satisfies the equation \(z^2 - 4iz^* + 11 = 0\). Given that \(\text{Re}(z) > 0\), find \(z\) in the form \(a + bi\), where \(a\) and \(b\) are real numbers. [4]
SPS SPS FM Pure 2025 February Q1
4 marks Standard +0.3
The complex number \(z\) satisfies the equation \(z^2 - 4iz* + 11 = 0\). Given that \(\text{Re}(z) > 0\), find \(z\) in the form \(a + bi\), where \(a\) and \(b\) are real numbers. [4]