What is the answer to (x^2+6)+(2x)i=15+6i ?

The answer to (x^2+6)+(2x)i=15+6i is x=3,x=-3-2i

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(x^2+6)+(2x)i=15+6i

(x^{2} + 6) + (2 x) i = 15 + 6 i

x = 3, x = - 3 - 2 i

Solution steps

(x^{2} + 6) + (2 x) i = 15 + 6 i

Substitute

x = a + bi

(a + bi)^{2} + 6 + 2 (a + bi) i = 15 + 6 i

Expand

(a + bi)^{2} + 6 + 2 (a + bi) i : (a^{2} - b^{2} + 6 - 2 b) + i (2 a + 2 ab)

(a^{2} - b^{2} + 6 - 2 b) + i (2 a + 2 ab) = 15 + 6 i

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

[a^{2} - b^{2} + 6 - 2 b = 15 2 a + 2 ab = 6]

[a^{2} - b^{2} + 6 - 2 b = 15 2 a + 2 ab = 6] : (a = 3, a = - 3, b = 0 b = - 2)

Substitute back

x = a + bi

x = 3, x = - 3 - 2 i

z^{2} - (1 + 6 i) z - 10 = 0

\frac{48 - 12 i}{x} = 16 - 4 i

y, (2 x - y) - 12 i = 12 + 6

x + 3 y + (2 x - 3 y - 9) i = 0

a + bi + \frac{1}{a + bi} = 1

What is the answer to (x^2+6)+(2x)i=15+6i ?
The answer to (x^2+6)+(2x)i=15+6i is x=3,x=-3-2i