What is the answer to zz+2z=12+6i ?

The answer to zz+2z=12+6i is z=2.69579…+0.81173…i,z=-4.69579…-0.81173…i

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zz+2z=12+6i

zz + 2 z = 12 + 6 i

z = 2.69579 \dots + 0.81173 \dots i, z = - 4.69579 \dots - 0.81173 \dots i

Solution steps

zz + 2 z = 12 + 6 i

Substitute

z = x + y i

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

Expand

(x + y i) (x + y i) + 2 (x + y i) : (x^{2} - y^{2} + 2 x) + i (2 y + 2 x y)

(x^{2} - y^{2} + 2 x) + i (2 y + 2 x y) = 12 + 6 i

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

[x^{2} - y^{2} + 2 x = 12 2 y + 2 x y = 6]

[x^{2} - y^{2} + 2 x = 12 2 y + 2 x y = 6] : (x = 2.69579 \dots, x = - 4.69579 \dots, y = 0.81173 \dots y = - 0.81173 \dots)

Substitute back

z = x + y i

z = 2.69579 \dots + 0.81173 \dots i, z = - 4.69579 \dots - 0.81173 \dots i

9 + (x + 4 y) i = x + 5 i

i (x) = e x^{2} + 2 π

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

a, a + bi = a + c i

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

What is the answer to zz+2z=12+6i ?
The answer to zz+2z=12+6i is z=2.69579…+0.81173…i,z=-4.69579…-0.81173…i