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.

7 questions · Moderate -1.0

4.02g Conjugate pairs: real coefficient polynomials
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Edexcel FP1 2016 June Q4
7 marks Moderate -0.3
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
5 marks Easy -1.2
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
4 marks Easy -1.2
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
5 marks Moderate -0.8
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
4 marks Easy -1.2
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 \(b\).
Pre-U Pre-U 9794/1 2014 June Q5
4 marks Easy -1.2
5 A root of the equation \(z ^ { 2 } + p z + q = 0\) is \(3 + \mathrm { i }\), where \(p\) and \(q\) are real. Write down the other root of the equation and hence calculate the values of \(p\) and \(q\).
Edexcel FP1 Q4
5 marks Moderate -0.8
Given that \(2 + i\) is a root of the equation $$z^2 + bz + c = 0, \text{ where } b \text{ and } c \text{ are real constants,}$$
  1. write down the other root of the equation,
  2. find the value of \(b\) and the value of \(c\). [5]