Solve (x-1)x=x-1

$$( x - 1 ) x = x - 1$$

Solve for x

$$x=1$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-1 by x.

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

Subtract x from both sides.

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

Combine -x and -x to get -2x.

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

Add 1 to both sides.

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

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

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

Square -2.

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

Add 4 to -4.

$$x=\frac{-\left(-2\right)±\sqrt{0}}{2}$$

Take the square root of 0.

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

The opposite of -2 is 2.

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

Divide 2 by 2.

$$x=1$$

Steps for Completing the Square

Use the distributive property to multiply x-1 by x.

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

Subtract x from both sides.

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

Combine -x and -x to get -2x.

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

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Add -1 to 1.

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

Factor x^{2}-2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-1\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{0}$$

Simplify.

$$x-1=0 x-1=0$$

Add 1 to both sides of the equation.

$$x=1 x=1$$

The equation is now solved. Solutions are the same.

$$x=1$$