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Solve for x (complex solution)
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x^{2}+2x+1-3x=-2
Subtract 3x from both sides.
x^{2}-x+1=-2
Combine 2x and -3x to get -x.
x^{2}-x+1+2=0
Add 2 to both sides.
x^{2}-x+3=0
Add 1 and 2 to get 3.
x=\frac{-\left(-1\right)±\sqrt{1-4\times 3}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -1 for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-1\right)±\sqrt{1-12}}{2}
Multiply -4 times 3.
x=\frac{-\left(-1\right)±\sqrt{-11}}{2}
Add 1 to -12.
x=\frac{-\left(-1\right)±\sqrt{11}i}{2}
Take the square root of -11.
x=\frac{1±\sqrt{11}i}{2}
The opposite of -1 is 1.
x=\frac{1+\sqrt{11}i}{2}
Now solve the equation x=\frac{1±\sqrt{11}i}{2} when ± is plus. Add 1 to i\sqrt{11}.
x=\frac{-\sqrt{11}i+1}{2}
Now solve the equation x=\frac{1±\sqrt{11}i}{2} when ± is minus. Subtract i\sqrt{11} from 1.
x=\frac{1+\sqrt{11}i}{2} x=\frac{-\sqrt{11}i+1}{2}
The equation is now solved.
x^{2}+2x+1-3x=-2
Subtract 3x from both sides.
x^{2}-x+1=-2
Combine 2x and -3x to get -x.
x^{2}-x=-2-1
Subtract 1 from both sides.
x^{2}-x=-3
Subtract 1 from -2 to get -3.
x^{2}-x+\left(-\frac{1}{2}\right)^{2}=-3+\left(-\frac{1}{2}\right)^{2}
Divide -1, the coefficient of the x term, by 2 to get -\frac{1}{2}. Then add the square of -\frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-x+\frac{1}{4}=-3+\frac{1}{4}
Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.
x^{2}-x+\frac{1}{4}=-\frac{11}{4}
Add -3 to \frac{1}{4}.
\left(x-\frac{1}{2}\right)^{2}=-\frac{11}{4}
Factor x^{2}-x+\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.
\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{-\frac{11}{4}}
Take the square root of both sides of the equation.
x-\frac{1}{2}=\frac{\sqrt{11}i}{2} x-\frac{1}{2}=-\frac{\sqrt{11}i}{2}
Simplify.
x=\frac{1+\sqrt{11}i}{2} x=\frac{-\sqrt{11}i+1}{2}
Add \frac{1}{2} to both sides of the equation.