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

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

Solve for x

$$x=3$$

$$x=0$$

Steps Using Factoring

Subtract x^{2} from both sides.

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

Add 2x to both sides.

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

Combine x and 2x to get 3x.

$$3x-1-x^{2}=-1$$

Add 1 to both sides.

$$3x-1-x^{2}+1=0$$

Add -1 and 1 to get 0.

$$3x-x^{2}=0$$

Factor out x.

$$x\left(3-x\right)=0$$

To find equation solutions, solve x=0 and 3-x=0.

$$x=0 x=3$$

Steps Using the Quadratic Formula

Subtract x^{2} from both sides.

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

Add 2x to both sides.

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

Combine x and 2x to get 3x.

$$3x-1-x^{2}=-1$$

Add 1 to both sides.

$$3x-1-x^{2}+1=0$$

Add -1 and 1 to get 0.

$$3x-x^{2}=0$$

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.

$$-x^{2}+3x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 3 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Take the square root of 3^{2}.

$$x=\frac{-3±3}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-3±3}{-2}$$

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

$$x=\frac{0}{-2}$$

Divide 0 by -2.

$$x=0$$

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

$$x=-\frac{6}{-2}$$

Divide -6 by -2.

$$x=3$$

The equation is now solved.

$$x=0 x=3$$

Steps for Completing the Square

Subtract x^{2} from both sides.

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

Add 2x to both sides.

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

Combine x and 2x to get 3x.

$$3x-1-x^{2}=-1$$

Add 1 to both sides.

$$3x-x^{2}=-1+1$$

Add -1 and 1 to get 0.

$$3x-x^{2}=0$$

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.

$$-x^{2}+3x=0$$

Divide both sides by -1.

$$\frac{-x^{2}+3x}{-1}=\frac{0}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{3}{-1}x=\frac{0}{-1}$$

Divide 3 by -1.

$$x^{2}-3x=\frac{0}{-1}$$

Divide 0 by -1.

$$x^{2}-3x=0$$

Divide -3, the coefficient of the x term, by 2 to get -\frac{3}{2}. Then add the square of -\frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-3x+\frac{9}{4}=\frac{9}{4}$$

Factor x^{2}-3x+\frac{9}{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}.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{9}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Simplify.

$$x-\frac{3}{2}=\frac{3}{2} x-\frac{3}{2}=-\frac{3}{2}$$

Add \frac{3}{2} to both sides of the equation.

$$x=3 x=0$$