Solve x=x^2+1

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

Solve for x (complex solution)

$$x=\frac{-\sqrt{3}i+1}{2}\approx 0.5-0.866025404i$$

$$x=\frac{1+\sqrt{3}i}{2}\approx 0.5+0.866025404i$$

Steps Using the Quadratic Formula

Subtract x^{2} from both sides.

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

Subtract 1 from both sides.

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

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

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

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

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

Square 1.

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

Multiply -4 times -1.

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

Multiply 4 times -1.

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

Add 1 to -4.

$$x=\frac{-1±\sqrt{-3}}{2\left(-1\right)}$$

Take the square root of -3.

$$x=\frac{-1±\sqrt{3}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-1±\sqrt{3}i}{-2}$$

Now solve the equation x=\frac{-1±\sqrt{3}i}{-2} when ± is plus. Add -1 to i\sqrt{3}.

$$x=\frac{-1+\sqrt{3}i}{-2}$$

Divide -1+i\sqrt{3} by -2.

$$x=\frac{-\sqrt{3}i+1}{2}$$

Now solve the equation x=\frac{-1±\sqrt{3}i}{-2} when ± is minus. Subtract i\sqrt{3} from -1.

$$x=\frac{-\sqrt{3}i-1}{-2}$$

Divide -1-i\sqrt{3} by -2.

$$x=\frac{1+\sqrt{3}i}{2}$$

The equation is now solved.

$$x=\frac{-\sqrt{3}i+1}{2} x=\frac{1+\sqrt{3}i}{2}$$

Steps for Completing the Square

Subtract x^{2} from both sides.

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

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

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

Divide both sides by -1.

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

Dividing by -1 undoes the multiplication by -1.

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

Divide 1 by -1.

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

Divide 1 by -1.

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

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.

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

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

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

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

$$x^{2}-x+\frac{1}{4}=-\frac{3}{4}$$

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}.

$$\left(x-\frac{1}{2}\right)^{2}=-\frac{3}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{-\frac{3}{4}}$$

Simplify.

$$x-\frac{1}{2}=\frac{\sqrt{3}i}{2} x-\frac{1}{2}=-\frac{\sqrt{3}i}{2}$$

Add \frac{1}{2} to both sides of the equation.

$$x=\frac{1+\sqrt{3}i}{2} x=\frac{-\sqrt{3}i+1}{2}$$