What is the answer to x^2+5x+2x=1-5i ?

The answer to x^2+5x+2x=1-5i is x=0.20216…-0.67528…i,x=-7.20216…+0.67528…i

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x^2+5x+2x=1-5i

x^{2} + 5 x + 2 x = 1 - 5 i

x = 0.20216 \dots - 0.67528 \dots i, x = - 7.20216 \dots + 0.67528 \dots i

Solution steps

x^{2} + 5 x + 2 x = 1 - 5 i

Substitute

x = a + bi

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

Expand

(a + bi)^{2} + 5 (a + bi) + 2 (a + bi) : (a^{2} - b^{2} + 7 a) + i (7 b + 2 ab)

(a^{2} - b^{2} + 7 a) + i (7 b + 2 ab) = 1 - 5 i

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

[a^{2} - b^{2} + 7 a = 1 7 b + 2 ab = - 5]

[a^{2} - b^{2} + 7 a = 1 7 b + 2 ab = - 5] : (a = 0.20216 \dots, a = - 7.20216 \dots, b = - 0.67528 \dots b = 0.67528 \dots)

Substitute back

x = a + bi

x = 0.20216 \dots - 0.67528 \dots i, x = - 7.20216 \dots + 0.67528 \dots i

3 (x - 2) = 2 y i - (1 - i), y + 2 = x - 3 i

a, z = (a + bi) (a - bi)

\frac{Z - i}{Z} = 2 + i

5 x + 3 i = 15 + y i

i (t) = t^{2} - 5 t + 6

What is the answer to x^2+5x+2x=1-5i ?
The answer to x^2+5x+2x=1-5i is x=0.20216…-0.67528…i,x=-7.20216…+0.67528…i