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.

8 questions

CAIE P3 2004 June Q8
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
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
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
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
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
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.
Edexcel FP1 Q18
18. The complex number \(z = a + \mathrm { i } b\), where \(a\) and \(b\) are real numbers, satisfies the equation $$z ^ { 2 } + 16 - 30 i = 0$$
  1. Show that \(a b = 15\).
  2. Write down a second equation in \(a\) and \(b\) and hence find the roots of $$z ^ { 2 } + 16 - 30 i = 0$$
OCR Further Pure Core AS 2021 November Q6
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 \(\mathrm { a } + \mathrm { bi }\).