Solve x^2+2x+1-x=-1

$${ x }^{ 2 } +2x+1-x = -1$$

Solve for x (complex solution)

$$x=\frac{-1+\sqrt{7}i}{2}\approx -0.5+1.322875656i$$

$$x=\frac{-\sqrt{7}i-1}{2}\approx -0.5-1.322875656i$$

Steps Using the Quadratic Formula

Combine 2x and -x to get x.

$$x^{2}+x+1=-1$$

Add 1 to both sides.

$$x^{2}+x+1+1=0$$

Add 1 and 1 to get 2.

$$x^{2}+x+2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 1 for b, and 2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\times 2}}{2}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\times 2}}{2}$$

Multiply -4 times 2.

$$x=\frac{-1±\sqrt{1-8}}{2}$$

Add 1 to -8.

$$x=\frac{-1±\sqrt{-7}}{2}$$

Take the square root of -7.

$$x=\frac{-1±\sqrt{7}i}{2}$$

Now solve the equation x=\frac{-1±\sqrt{7}i}{2} when ± is plus. Add -1 to i\sqrt{7}.

$$x=\frac{-1+\sqrt{7}i}{2}$$

Now solve the equation x=\frac{-1±\sqrt{7}i}{2} when ± is minus. Subtract i\sqrt{7} from -1.

$$x=\frac{-\sqrt{7}i-1}{2}$$

The equation is now solved.

$$x=\frac{-1+\sqrt{7}i}{2} x=\frac{-\sqrt{7}i-1}{2}$$

Steps for Completing the Square

Combine 2x and -x to get x.

$$x^{2}+x+1=-1$$

Subtract 1 from both sides.

$$x^{2}+x=-1-1$$

Subtract 1 from -1 to get -2.

$$x^{2}+x=-2$$

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.

$$x^{2}+x+\left(\frac{1}{2}\right)^{2}=-2+\left(\frac{1}{2}\right)^{2}$$

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+x+\frac{1}{4}=-2+\frac{1}{4}$$

Add -2 to \frac{1}{4}.

$$x^{2}+x+\frac{1}{4}=-\frac{7}{4}$$

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}.

$$\left(x+\frac{1}{2}\right)^{2}=-\frac{7}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{-\frac{7}{4}}$$

Simplify.

$$x+\frac{1}{2}=\frac{\sqrt{7}i}{2} x+\frac{1}{2}=-\frac{\sqrt{7}i}{2}$$

Subtract \frac{1}{2} from both sides of the equation.

$$x=\frac{-1+\sqrt{7}i}{2} x=\frac{-\sqrt{7}i-1}{2}$$