$$(3+4i)^{2}-2(x-yi)=x+yi$$
$x=\frac{iy}{3}+\left(-\frac{7}{3}+8i\right)$
$$-7+24i-2\left(x-yi\right)=x+yi$$
$$-7+24i-2\left(x-yi\right)-x=yi$$
$$-7+24i-2\left(x-iy\right)-x=yi$$
$$-7+24i-2x+2iy-x=yi$$
$$-7+24i-3x+2iy=yi$$
$$-3x+2iy=yi-\left(-7+24i\right)$$
$$-3x=yi-\left(-7+24i\right)-2iy$$
$$-3x=7-24i-iy$$
$$\frac{-3x}{-3}=\frac{7-24i-iy}{-3}$$
$$x=\frac{7-24i-iy}{-3}$$
$$x=\frac{iy}{3}+\left(-\frac{7}{3}+8i\right)$$
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$y=-24-7i-3ix$
$$-7+24i-2\left(x-yi\right)-yi=x$$
$$-7+24i-2\left(x-iy\right)-yi=x$$
$$-7+24i-2x+2iy-yi=x$$
$$-7+24i-2x+2iy-iy=x$$
$$-7+24i-2x+iy=x$$
$$-2x+iy=x-\left(-7+24i\right)$$
$$iy=x-\left(-7+24i\right)+2x$$
$$iy=3x+\left(7-24i\right)$$
$$\frac{iy}{i}=\frac{3x+\left(7-24i\right)}{i}$$
$$y=\frac{3x+\left(7-24i\right)}{i}$$
$$y=-24-7i-3ix$$
