Standard quadratic with real coefficients

Quadratic equations with real coefficients only, solved using the quadratic formula or completing the square to obtain complex roots in Cartesian form.

9 questions · Moderate -0.7

4.02i Quadratic equations: with complex roots
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CAIE P3 2004 June Q8
7 marks Moderate -0.3
8
  1. Find the roots of the equation \(z ^ { 2 } - z + 1 = 0\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. Obtain the modulus and argument of each root.
  3. Show that each root also satisfies the equation \(z ^ { 3 } = - 1\).
Edexcel FP1 2011 June Q2
8 marks Moderate -0.8
2. $$z _ { 1 } = - 2 + \mathrm { i }$$
  1. Find the modulus of \(z _ { 1 }\).
  2. Find, in radians, the argument of \(z _ { 1 }\), giving your answer to 2 decimal places. The solutions to the quadratic equation $$z ^ { 2 } - 10 z + 28 = 0$$ are \(z _ { 2 }\) and \(z _ { 3 }\).
  3. Find \(z _ { 2 }\) and \(z _ { 3 }\), giving your answers in the form \(p \pm i \sqrt { } q\), where \(p\) and \(q\) are integers.
  4. Show, on an Argand diagram, the points representing your complex numbers \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\).
OCR MEI FP1 2007 January Q2
6 marks Easy -1.2
2
  1. Find the roots of the quadratic equation \(z ^ { 2 } - 4 z + 7 = 0\), simplifying your answers as far as possible.
  2. Represent these roots on an Argand diagram.
OCR MEI FP1 2009 January Q1
5 marks Easy -1.2
1
  1. Find the roots of the quadratic equation \(z ^ { 2 } - 6 z + 10 = 0\) in the form \(a + b \mathrm { j }\).
  2. Express these roots in modulus-argument form.
OCR MEI FP1 2015 June Q2
5 marks Moderate -0.8
2 Find the roots of the quadratic equation \(z ^ { 2 } - 4 z + 13 = 0\).
Find the modulus and argument of each root.
OCR Further Pure Core 2 2020 November Q1
5 marks Moderate -0.3
1 In this question you must show detailed reasoning.
Solve the equation \(4 z ^ { 2 } - 20 z + 169 = 0\). Give your answers in modulus-argument form.
AQA FP1 2013 January Q2
9 marks Moderate -0.3
2
  1. Solve the equation \(w ^ { 2 } + 6 w + 34 = 0\), giving your answers in the form \(p + q \mathrm { i }\), where \(p\) and \(q\) are integers.
  2. It is given that \(z = \mathrm { i } ( 1 + \mathrm { i } ) ( 2 + \mathrm { i } )\).
    1. Express \(z\) in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are integers.
    2. Find integers \(m\) and \(n\) such that \(z + m z ^ { * } = n \mathrm { i }\).
OCR Further Pure Core AS 2021 November Q6
6 marks Moderate -0.8
6 In this question you must show detailed reasoning.
  1. Solve the equation \(2 z ^ { 2 } - 10 z + 25 = 0\) giving your answers in the form \(\mathrm { a } + \mathrm { bi }\).
  2. Solve the equation \(3 \omega - 2 = \mathrm { i } ( 5 + 2 \omega )\) giving your answer in the form \(\mathrm { a } + \mathrm { bi }\).
Edexcel FP1 Q35
4 marks Moderate -0.8
  1. Find the roots of the equation \(z^2 + 2z + 17 = 0\), giving your answers in the form \(a + ib\), where \(a\) and \(b\) are integers. [3]
  2. Show these roots on an Argand diagram. [1]