4.02i Quadratic equations: with complex roots

8

1. 1. It is given that \(\alpha\) and \(\beta\) are the roots of the equation $$x ^ { 2 } - 2 x + 4 = 0$$ Without solving this equation, show that \(\alpha ^ { 3 }\) and \(\beta ^ { 3 }\) are the roots of the equation $$x ^ { 2 } + 16 x + 64 = 0$$ (6 marks)
   2. State, giving a reason, whether the roots of the equation $$x ^ { 2 } + 16 x + 64 = 0$$ are real and equal, real and distinct, or non-real.
2. Solve the equation $$x ^ { 2 } - 2 x + 4 = 0$$
3. Use your answers to parts (a) and (b) to show that $$( 1 + \mathrm { i } \sqrt { 3 } ) ^ { 3 } = ( 1 - \mathrm { i } \sqrt { 3 } ) ^ { 3 }$$

3 The cubic equation $$z ^ { 3 } + 2 ( 1 - \mathrm { i } ) z ^ { 2 } + 32 ( 1 + \mathrm { i } ) = 0$$ has roots \(\alpha , \beta\) and \(\gamma\).

1. It is given that \(\alpha\) is of the form \(k \mathrm { i }\), where \(k\) is real. By substituting \(z = k \mathrm { i }\) into the equation, show that \(k = 4\).
2. Given that \(\beta = - 4\), find the value of \(\gamma\).

5 The cubic equation $$z ^ { 3 } - 4 \mathrm { i } z ^ { 2 } + q z - ( 4 - 2 \mathrm { i } ) = 0$$ where \(q\) is a complex number, has roots \(\alpha , \beta\) and \(\gamma\).

1. Write down the value of:
   1. \(\alpha + \beta + \gamma\);
   2. \(\alpha \beta \gamma\).
2. Given that \(\alpha = \beta + \gamma\), show that:
   1. \(\alpha = 2 \mathrm { i }\);
   2. \(\quad \beta \gamma = - ( 1 + 2 \mathrm { i } )\);
   3. \(\quad q = - ( 5 + 2 \mathrm { i } )\).
3. Show that \(\beta\) and \(\gamma\) are the roots of the equation $$z ^ { 2 } - 2 \mathrm { i } z - ( 1 + 2 \mathrm { i } ) = 0$$
4. Given that \(\beta\) is real, find \(\beta\) and \(\gamma\).

1

1. Find the roots of the equation \(m ^ { 2 } + 2 m + 2 = 0\) in the form \(a + i b\).
   (2 marks)
2. 1. Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y = 4 x$$
   2. Hence express \(y\) in terms of \(x\), given that \(y = 1\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2\) when \(x = 0\).

4 In this question you must show detailed reasoning. You are given that \(\mathrm { f } ( \mathrm { z } ) = 4 \mathrm { z } ^ { 4 } - 12 \mathrm { z } ^ { 3 } + 41 \mathrm { z } ^ { 2 } - 128 \mathrm { z } + 185\) and that \(2 + \mathrm { i }\) is a root of the equation \(f ( z ) = 0\).

1. Express \(\mathrm { f } ( \mathrm { z } )\) as the product of two quadratic factors with integer coefficients.
2. Solve \(f ( z ) = 0\). Two loci on an Argand diagram are defined by \(C _ { 1 } = \left\{ z : | z | = r _ { 1 } \right\}\) and \(C _ { 2 } = \left\{ z : | z | = r _ { 2 } \right\}\) where \(r _ { 1 } > r _ { 2 }\). You are given that two of the points representing the roots of \(\mathrm { f } ( \mathrm { z } ) = 0\) are on \(C _ { 1 }\) and two are on \(C _ { 2 } . R\) is the region on the Argand diagram between \(C _ { 1 }\) and \(C _ { 2 }\).
3. Find the exact area of \(R\).
4. \(\omega\) is the sum of all the roots of \(\mathrm { f } ( \mathrm { z } ) = 0\). Determine whether or not the point on the Argand diagram which represents \(\omega\) lies in \(R\).

3 In this question you must show detailed reasoning.
The equation \(x ^ { 4 } - 7 x ^ { 3 } - 2 x ^ { 2 } + 218 x - 1428 = 0\) has a root \(3 - 5 i\).
Find the other three roots of this equation.

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 \(\mathrm { a } + \mathrm { bi }\).

2 In this question you must show detailed reasoning. You are given that \(\mathrm { f } ( z ) = 4 z ^ { 4 } - 12 z ^ { 3 } + 41 z ^ { 2 } - 128 z + 185\) and that \(2 + \mathrm { i }\) is a root of the equation \(\mathrm { f } ( z ) = 0\).

1. Express \(\mathrm { f } ( z )\) as the product of two quadratic factors with integer coefficients.
2. Solve \(\mathrm { f } ( z ) = 0\). Two loci on an Argand diagram are defined by \(C _ { 1 } = \left\{ z : | z | = r _ { 1 } \right\}\) and \(C _ { 2 } = \left\{ z : | z | = r _ { 2 } \right\}\) where \(r _ { 1 } > r _ { 2 }\). You are given that two of the points representing the roots of \(\mathrm { f } ( z ) = 0\) are on \(C _ { 1 }\) and two are on \(C _ { 2 } \cdot R\) is the region on the Argand diagram between \(C _ { 1 }\) and \(C _ { 2 }\).
3. Find the exact area of \(R\).
4. \(\omega\) is the sum of all the roots of \(\mathrm { f } ( z ) = 0\). Determine whether or not the point on the Argand diagram which represents \(\omega\) lies in \(R\).

4

1. Verify that \(z = - 1\) is a root of the equation \(z ^ { 3 } + 5 z ^ { 2 } + 9 z + 5 = 0\).
2. Find the two complex roots of the equation \(z ^ { 3 } + 5 z ^ { 2 } + 9 z + 5 = 0\).
3. Show all three roots on an Argand diagram.

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 \(q\).

13 The complex number \(w\) has modulus 1. It is given that $$w ^ { 2 } - \frac { 2 } { w } + k \mathrm { i } = 0$$ where \(k\) is a positive real constant.

1. Show that \(k = ( 3 - \sqrt { 3 } ) \sqrt { \frac { 1 } { 2 } \sqrt { 3 } }\).
2. Prove that at least one of the remaining two roots of the equation \(z ^ { 2 } - \frac { 2 } { z } + k i = 0\) has modulus greater than 1 .

9 Solve the equation \(z ^ { 3 } + 6 z - 20 = 0\). Find the modulus and argument of each root and illustrate the roots on an Argand diagram.

9

1. Show that \(z = ( 1 + \mathrm { i } )\) is a root of the cubic equation \(3 z ^ { 3 } - 8 z ^ { 2 } + 10 z - 4 = 0\).
2. Show that the equation \(3 z ^ { 3 } - 8 z ^ { 2 } + 10 z - 4 = 0\) has a quadratic factor with real coefficients and hence solve this equation completely.

Throughout this question the use of a calculator is not permitted. The complex number \(2 - \mathrm{i}\) is denoted by \(u\).

1. It is given that \(u\) is a root of the equation \(x^3 + ax^2 - 3x + b = 0\), where the constants \(a\) and \(b\) are real. Find the values of \(a\) and \(b\). [4]
2. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying both the inequalities \(|z - u| < 1\) and \(|z| < |z + \mathrm{i}|\). [4]

Find the sum of the squares of the roots of the equation $$x^3 + x + 12 = 0,$$ and deduce that only one of the roots is real. [4] The real root of the equation is denoted by \(\alpha\). Prove that \(-3 < \alpha < -2\), and hence prove that the modulus of each of the other roots lies between 2 and \(\sqrt{6}\). [5]

The equation $$x^4 + Ax^3 + Bx^2 + Cx + 225 = 0$$ where \(A\), \(B\) and \(C\) are real constants, has

• a complex root \(4 + 3\text{i}\)
• a repeated positive real root

1. Write down the other complex root of this equation. [1]
2. Hence determine a quadratic factor of \(x^4 + Ax^3 + Bx^2 + Cx + 225\) [2]
3. Deduce the real root of the equation. [2]
4. Hence determine the value of each of the constants \(A\), \(B\) and \(C\) [3]

$$f(x) = (4x^2 + 9)(x^2 - 2x + 5)$$

1. Find the four roots of \(f(x) = 0\) [4]
2. Show the four roots of \(f(x) = 0\) on a single Argand diagram. [2]

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]

The complex number \(z = a + ib\), where \(a\) and \(b\) are real numbers, satisfies the equation $$z^2 + 16 - 30i = 0.$$

1. Show that \(ab = 15\). [2]
2. Write down a second equation in \(a\) and \(b\) and hence find the roots of \(z^2 + 16 - 30i = 0\). [4]

Given that \(-2\) is a root of the equation \(z^3 + 6z + 20 = 0\),

1. Find the other two roots of the equation, [3]
2. show, on a single Argand diagram, the three points representing the roots of the equation, [1]
3. prove that these three points are the vertices of a right-angled triangle. [2]

Given that \(x = -\frac{1}{2}\) is the real solution of the equation $$2x^3 - 11x^2 + 14x + 10 = 0,$$ find the two complex solutions of this equation. [6]