Quadratic from one complex root

Given one complex root of a quadratic equation with real coefficients, find the other root (conjugate) and/or the real coefficients of the equation.

8 questions

Edexcel FP1 2016 June Q4
4. $$z = \frac { 4 } { 1 + \mathrm { i } }$$ Find, in the form \(a + \mathrm { i } b\) where \(a , b \in \mathbb { R }\)
  1. \(Z\)
  2. \(z ^ { 2 }\) Given that \(z\) is a complex root of the quadratic equation \(x ^ { 2 } + p x + q = 0\), where \(p\) and \(q\) are real integers,
  3. find the value of \(p\) and the value of \(q\).
OCR FP1 2006 June Q3
3 One root of the quadratic equation \(x ^ { 2 } + p x + q = 0\), where \(p\) and \(q\) are real, is the complex number 2-3i.
  1. Write down the other root.
  2. Find the values of \(p\) and \(q\).
OCR FP1 2012 June Q3
3 One root of the quadratic equation \(x ^ { 2 } + a x + b = 0\), where \(a\) and \(b\) are real, is the complex number \(4 - 3 \mathrm { i }\). Find the values of \(a\) and \(b\).
AQA FP1 2009 January Q2
2 The complex number \(2 + 3 \mathrm { i }\) is a root of the quadratic equation $$x ^ { 2 } + b x + c = 0$$ where \(b\) and \(c\) are real numbers.
  1. Write down the other root of this equation.
  2. Find the values of \(b\) and \(c\).
OCR MEI Further Pure Core AS 2024 June Q1
1 The quadratic equation \(\mathrm { x } ^ { 2 } + \mathrm { ax } + \mathrm { b } = 0\), where \(a\) and \(b\) are real constants, has a root 2-3.
  1. Write down the other root.
  2. Hence or otherwise determine the values of \(a\) and \(b\).
CAIE P3 2017 June Q6
6 Throughout this question the use of a calculator is not permitted. The complex number \(2 - \mathrm { i }\) is denoted by \(u\).
  1. It is given that \(u\) is a root of the equation \(x ^ { 3 } + a x ^ { 2 } - 3 x + b = 0\), where the constants \(a\) and \(b\) are real. Find the values of \(a\) and \(b\).
  2. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying both the inequalities \(| z - u | < 1\) and \(| z | < | z + \mathrm { i } |\).
Edexcel FP1 Q4
4. Given that \(2 + \mathrm { i }\) is a root of the equation $$z ^ { 2 } + b z + c = 0 , \text { where } b \text { and } c \text { are real constants, }$$
  1. write down the other root of the equation,
  2. find the value of \(b\) and the value of \(c\).
AQA Further Paper 1 2021 June Q2
2 Given that \(z = 1 - 3 \mathrm { i }\) is one root of the equation \(z ^ { 2 } + p z + r = 0\), where \(p\) and \(r\) are real, find the value of \(r\). Circle your answer.
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