4.02i Quadratic equations: with complex roots

6 Solve the quadratic equation \(( 1 - 3 \mathrm { i } ) z ^ { 2 } - ( 2 + \mathrm { i } ) z + \mathrm { i } = 0\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.

4 Solve the quadratic equation \(( 3 + \mathrm { i } ) w ^ { 2 } - 2 w + 3 - \mathrm { i } = 0\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.

1

1. Find \(a\) and \(b\) such that $$z ^ { 8 } - i z ^ { 5 } - z ^ { 3 } + i = \left( z ^ { 5 } - a \right) \left( z ^ { 3 } - b \right) .$$
2. Hence find the roots of $$z ^ { 8 } - i z ^ { 5 } - z ^ { 3 } + i = 0$$ giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(0 \leqslant \theta < 2 \pi\).

3. Given that \(z = x + \mathrm { i } y\), where \(x\) and \(y\) are real numbers, solve the equation $$( z - 2 i ) \left( z ^ { * } - 2 i \right) = 21 - 12 i$$ where \(z ^ { * }\) is the complex conjugate of \(z\).

3. $$f ( x ) = x ^ { 4 } + 2 x ^ { 3 } + 26 x ^ { 2 } + 32 x + 160$$ Given that \(x = - 1 + 3 \mathrm { i }\) is a root of the equation \(\mathrm { f } ( x ) = 0\), use algebra to find the three other roots of \(\mathrm { f } ( x ) = 0\) (Solutions based entirely on graphical or numerical methods are not acceptable.)

2. $$f ( z ) = z ^ { 4 } - 6 z ^ { 3 } + 38 z ^ { 2 } - 94 z + 221$$

1. Given that \(z = 2 + 3 i\) is a root of the equation \(f ( z ) = 0\), use algebra to find the three other roots of \(f ( z ) = 0\)
2. Show the four roots of \(\mathrm { f } ( \mathrm { z } ) = 0\) on a single Argand diagram.

1. Given that \(x = \frac { 3 } { 8 } + \frac { \sqrt { 71 } } { 8 } \mathrm { i }\) is a root of the equation

$$4 x ^ { 3 } - 19 x ^ { 2 } + p x + q = 0$$

1. write down the other complex root of the equation. Given that \(x = 4\) is also a root of the equation,
2. find the value of \(p\) and the value of \(q\).

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

$$\mathrm { f } ( z ) = 4 z ^ { 3 } + p z ^ { 2 } - 24 z + 108$$ where \(p\) is a constant.
Given that - 3 is a root of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)

1. determine the value of \(p\)
2. using algebra, solve \(\mathrm { f } ( \mathrm { z } ) = 0\) completely, giving the roots in simplest form,
3. determine the modulus of the complex roots of \(\mathrm { f } ( \mathrm { z } ) = 0\)
4. show the roots of \(\mathrm { f } ( \mathrm { z } ) = 0\) on a single Argand diagram.

1. Given that

$$2 z ^ { 3 } - 5 z ^ { 2 } + 7 z - 6 \equiv ( 2 z - 3 ) \left( z ^ { 2 } + a z + b \right)$$ where \(a\) and \(b\) are real constants,

1. find the value of \(a\) and the value of \(b\).
2. Given that \(z\) is a complex number, find the three exact roots of the equation $$2 z ^ { 3 } - 5 z ^ { 2 } + 7 z - 6 = 0$$

8. $$f ( z ) = z ^ { 4 } + 6 z ^ { 3 } + 76 z ^ { 2 } + a z + b$$ where \(a\) and \(b\) are real constants. Given that \(- 3 + 8 \mathrm { i }\) is a complex root of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)

1. write down another complex root of this equation.
2. Hence, or otherwise, find the other roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)
3. Show on a single Argand diagram all four roots of the equation \(f ( z ) = 0\)

4. $$f ( z ) = 2 z ^ { 3 } - z ^ { 2 } + a z + b$$ where \(a\) and \(b\) are integers. The complex number \(- 1 - 3 \mathrm { i }\) is a root of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)

1. Write down another complex root of this equation.
2. Determine the value of \(a\) and the value of \(b\).
3. Show all the roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\) on a single Argand diagram.

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4. Given that \(2 - 4 \mathrm { i }\) is a root of the equation $$z ^ { 2 } + p z + q = 0 ,$$ where \(p\) and \(q\) are real constants,

1. write down the other root of the equation,
2. find the value of \(p\) and the value of \(q\).

5. The roots of the equation $$z ^ { 3 } - 8 z ^ { 2 } + 22 z - 20 = 0$$ are \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\).

1. Given that \(z _ { 1 } = 3 + \mathrm { i }\), find \(z _ { 2 }\) and \(z _ { 3 }\).
2. Show, on a single Argand diagram, the points representing \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\).

5. $$f ( x ) = \left( 4 x ^ { 2 } + 9 \right) \left( x ^ { 2 } - 6 x + 34 \right)$$

1. Find the four roots of \(\mathrm { f } ( x ) = 0\) Give your answers in the form \(x = p + \mathrm { i } q\), where \(p\) and \(q\) are real.
2. Show these four roots on a single Argand diagram.

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 \(z _ { 3 }\).

1. $$f ( z ) = 2 z ^ { 3 } - 4 z ^ { 2 } + 15 z - 13$$ Given that \(\mathrm { f } ( z ) \equiv ( z - 1 ) \left( 2 z ^ { 2 } + a z + b \right)\), where \(a\) and \(b\) are real constants,

1. find the value of \(a\) and the value of \(b\).
2. Hence use algebra to find the three roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)

7. The quadratic equation $$z ^ { 2 } + 10 z + 169 = 0$$ has complex roots \(z _ { 1 }\) and \(z _ { 2 }\).

1. Find each of these roots in the form \(a + b \mathrm { i }\).
2. Find the modulus and argument of \(z _ { 1 }\) and of \(z _ { 2 }\). Give the arguments in radians to 3 significant figures.
3. Illustrate the two roots on a single Argand diagram.
4. Find the value of \(\left| z _ { 1 } - z _ { 2 } \right|\).

8. $$\mathrm { f } ( x ) \equiv 2 x ^ { 3 } - 5 x ^ { 2 } + p x - 5 , p \in \mathbb { R }$$ Given that \(1 - 2 \mathrm { i }\) is a complex solution of \(\mathrm { f } ( x ) = 0\),

1. write down the other complex solution of \(\mathrm { f } ( x ) = 0\),
2. solve the equation \(\mathrm { f } ( x ) = 0\),
3. find the value of \(p\).

3

1. Solve the equation \(z ^ { 2 } - 6 z + 36 = 0\), and give your answers in the form \(r ( \cos \theta \pm \mathrm { i } \sin \theta )\), where \(r > 0\) and \(0 \leqslant \theta \leqslant \pi\).
2. Given that \(Z\) is either of the roots found in part (i), deduce the exact value of \(Z ^ { - 3 }\).

5

1. Verify that \(z ^ { 3 } - 8 = ( z - 2 ) \left( z ^ { 2 } + 2 z + 4 \right)\).
2. Solve the quadratic equation \(z ^ { 2 } + 2 z + 4 = 0\), giving your answers exactly in the form \(x + \mathrm { i } y\). Show clearly how you obtain your answers.
3. Show on an Argand diagram the roots of the cubic equation \(z ^ { 3 } - 8 = 0\).

10

1. Use an algebraic method to find the square roots of the complex number \(16 + 30 \mathrm { i }\).
2. Use your answers to part (i) to solve the equation \(z ^ { 2 } - 2 z - ( 15 + 30 \mathrm { i } ) = 0\), giving your answers in the form \(x + \mathrm { i } y\).

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.

5. $$f ( x ) = \left( 9 x ^ { 2 } + d \right) \left( x ^ { 2 } - 8 x + ( 10 d + 1 ) \right)$$ where \(d\) is a positive constant.

1. Find the four roots of \(\mathrm { f } ( x )\) giving your answers in terms of \(d\). Given \(d = 4\)
2. Express these four roots in the form \(a + \mathrm { i } b\), where \(a , b \in \mathbb { R }\).
3. Show these four roots on a single Argand diagram. \includegraphics[max width=\textwidth, alt={}, center]{d7689f4a-a41e-45be-911b-4a74e81997eb-21_2647_1840_118_111}

9

1. Show that \(\frac { 1 } { 2 r - 3 } - \frac { 1 } { 2 r + 1 } = \frac { 4 } { 4 r ^ { 2 } - 4 r - 3 }\).
2. Hence find an expression, in terms of \(n\), for $$\sum _ { r = 2 } ^ { n } \frac { 4 } { 4 r ^ { 2 } - 4 r - 3 }$$
3. Show that \(\sum _ { r = 2 } ^ { \infty } \frac { 4 } { 4 r ^ { 2 } - 4 r - 3 } = \frac { 4 } { 3 }\).
4. Use an algebraic method to find the square roots of the complex number \(2 + \mathrm { i } \sqrt { 5 }\). Give your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are exact real numbers.
5. Hence find, in the form \(x + \mathrm { i } y\) where \(x\) and \(y\) are exact real numbers, the roots of the equation $$z ^ { 4 } - 4 z ^ { 2 } + 9 = 0$$
6. Show, on an Argand diagram, the roots of the equation in part (ii).
7. Given that \(\alpha\) is the root of the equation in part (ii) such that \(0 < \arg \alpha < \frac { 1 } { 2 } \pi\), sketch on the same Argand diagram the locus given by \(| z - \alpha | = | z |\).

9 One root of the quadratic equation \(x ^ { 2 } + a x + b = 0\), where \(a\) and \(b\) are real, is \(16 - 30 \mathrm { i }\).

1. Write down the other root of the quadratic equation.
2. Find the values of \(a\) and \(b\).
3. Use an algebraic method to solve the quartic equation \(y ^ { 4 } + a y ^ { 2 } + b = 0\).