4.02h Square roots: of complex numbers

5 In this question you must show detailed reasoning.

1. Use an algebraic method to find the square roots of \(- 16 + 30 \mathrm { i }\).
2. By finding the cube of one of your answers to part (a) determine a cube root of \(\frac { - 99 + 5 i } { 4 }\). Give your answer in the form \(a + b \mathrm { i }\).

4 In this question you must show detailed reasoning.

1. Determine the square roots of 25 i in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(0 \leqslant \theta < 2 \pi\).
2. Illustrate the number 25i and its square roots on an Argand diagram.

1. $$f ( z ) = z ^ { 4 } + a z ^ { 3 } + b z ^ { 2 } + c z + d$$ where \(a , b , c\) and \(d\) are real constants.
Given that \(- 1 + 2 \mathrm { i }\) and \(3 - \mathrm { i }\) are two roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)

1. show all the roots of \(f ( z ) = 0\) on a single Argand diagram,
2. find the values of \(a , b , c\) and \(d\).

1 In this question you must show detailed reasoning.
Find the square roots of \(24 + 10 \mathrm { i }\), giving your answers in the form \(a + b \mathrm { i }\).

7 In this question you must show detailed reasoning.

1. Find the square roots of the number \(528 + 46 \mathrm { i }\) giving your answers in the form \(a + b \mathrm { i }\).
2. \(\quad 3 + 2 \mathrm { i }\) is a root of the equation \(x ^ { 3 } - a x + 78 = 0\), where \(a\) is a real number. Find the value of \(a\).

1 In this question you must show detailed reasoning.
Use an algebraic method to find the square roots of \(- 77 - 36 \mathrm { i }\).

2 In this question you must show detailed reasoning.

1. Determine the square roots of 25 i in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(0 \leqslant \theta < 2 \pi\).
2. Illustrate the number 25 i and its square roots on an Argand diagram.

The square roots of \(24 - 7i\) can be expressed in the Cartesian form \(x + iy\), where \(x\) and \(y\) are real and exact. By first forming a quartic equation in \(x\) or \(y\), find the square roots of \(24 - 7i\) in exact Cartesian form. [5]

The square roots of \(6 - 8i\) can be expressed in the Cartesian form \(x + iy\), where \(x\) and \(y\) are real and exact. By first forming a quartic equation in \(x\) or \(y\), find the square roots of \(6 - 8i\) in exact Cartesian form. [5]

Use an algebraic method to find the square roots of the complex number \(21 - 20i\). [6]

Use an algebraic method to find the square roots of the complex number \(21 - 20\text{i}\). [6]

The complex number \(z\), where \(0 < \arg z < \frac{1}{2}\pi\), is such that \(z^2 = 3 + 4\text{i}\).

1. Use an algebraic method to find \(z\). [5]
2. Show that \(z^3 = 2 + 11\text{i}\). [1]

The complex number \(w\) is the root of the equation $$w^6 - 4w^3 + 125 = 0$$ for which \(-\frac{1}{2}\pi < \arg w < 0\).

1. Find \(w\). [5]

In this question you must show detailed reasoning. Use an algebraic method to find the square roots of \(-77 - 36\text{i}\). [6]

Solve the quadratic equation \(x^2 - 4x - 1 - 12i = 0\) writing your solutions in the form \(a + bi\). [8]