Answer
$\displaystyle x=\frac{1}{2}\pm\frac{\sqrt{11}}{2}i$
Work Step by Step
We are given:
$x^{2}+3=x$
$x^{2}-x+3=0$
We solve using the quadratic formula ($a=1,\ b=-1,\ c=3$):
$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
$\displaystyle x=\frac{-(-1)\pm\sqrt{(-1)^{2}-4*1*3}}{2(1)}$
$\displaystyle x=\frac{-(-1)\pm\sqrt{1-12}}{2(1)}$
$\displaystyle x=\frac{+1\pm\sqrt{-11}}{2}$
$\displaystyle x=\frac{1\pm\sqrt{11}i}{2}$
$\displaystyle x=\frac{1}{2}\pm\frac{\sqrt{11}}{2}i$
