What is the answer to solvefor x,x^2=x(4+5i)+1 ?

The answer to solvefor x,x^2=x(4+5i)+1 is x=-0.10093…+0.12010…i,x=4.10093…+4.87989…i

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solvefor x,x^2=x(4+5i)+1

solve for

x, x^{2} = x (4 + 5 i) + 1

x = - 0.10093 \dots + 0.12010 \dots i, x = 4.10093 \dots + 4.87989 \dots i

Solution steps

x^{2} = x (4 + 5 i) + 1

Substitute

x = a + bi

(a + bi)^{2} = (a + bi) (4 + 5 i) + 1

Expand

(a^{2} - b^{2}) + 2 iab = (4 a - 5 b + 1) + i (4 b + 5 a)

Complex numbers can be equal only if their real and imaginary parts are equalRewrite as system of equations:

[a^{2} - b^{2} = 4 a - 5 b + 1 2 ab = 4 b + 5 a]

[a^{2} - b^{2} = 4 a - 5 b + 1 2 ab = 4 b + 5 a] : (a = - 0.10093 \dots, a = 4.10093 \dots, b = 0.12010 \dots b = 4.87989 \dots)

Substitute back

x = a + bi

x = - 0.10093 \dots + 0.12010 \dots i, x = 4.10093 \dots + 4.87989 \dots i

x + y i - \frac{1}{x + y i} = 1 + 4 i

4 z^{2} + (4 - 4 i) z + 5 - 2 i = 0

- 1 + 2 i = \frac{2 i}{( z - 1 )}

- x + y i + 7 = 6 i

What is the answer to solvefor x,x^2=x(4+5i)+1 ?
The answer to solvefor x,x^2=x(4+5i)+1 is x=-0.10093…+0.12010…i,x=4.10093…+4.87989…i