Solve 180*(x-2)-180(x-2)/x=180

$$180 \times (x-2)- \frac{ 180(x-2) }{ x } =180$$

Solve for x

$$x=\sqrt{2}+2\approx 3.414213562$$

$$x=2-\sqrt{2}\approx 0.585786438$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$180\left(x-2\right)x-180\left(x-2\right)=180x$$

Use the distributive property to multiply 180 by x-2.

$$\left(180x-360\right)x-180\left(x-2\right)=180x$$

Use the distributive property to multiply 180x-360 by x.

$$180x^{2}-360x-180\left(x-2\right)=180x$$

Use the distributive property to multiply -180 by x-2.

$$180x^{2}-360x-180x+360=180x$$

Combine -360x and -180x to get -540x.

$$180x^{2}-540x+360=180x$$

Subtract 180x from both sides.

$$180x^{2}-540x+360-180x=0$$

Combine -540x and -180x to get -720x.

$$180x^{2}-720x+360=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 180 for a, -720 for b, and 360 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-720\right)±\sqrt{\left(-720\right)^{2}-4\times 180\times 360}}{2\times 180}$$

Square -720.

$$x=\frac{-\left(-720\right)±\sqrt{518400-4\times 180\times 360}}{2\times 180}$$

Multiply -4 times 180.

$$x=\frac{-\left(-720\right)±\sqrt{518400-720\times 360}}{2\times 180}$$

Multiply -720 times 360.

$$x=\frac{-\left(-720\right)±\sqrt{518400-259200}}{2\times 180}$$

Add 518400 to -259200.

$$x=\frac{-\left(-720\right)±\sqrt{259200}}{2\times 180}$$

Take the square root of 259200.

$$x=\frac{-\left(-720\right)±360\sqrt{2}}{2\times 180}$$

The opposite of -720 is 720.

$$x=\frac{720±360\sqrt{2}}{2\times 180}$$

Multiply 2 times 180.

$$x=\frac{720±360\sqrt{2}}{360}$$

Now solve the equation x=\frac{720±360\sqrt{2}}{360} when ± is plus. Add 720 to 360\sqrt{2}.

$$x=\frac{360\sqrt{2}+720}{360}$$

Divide 720+360\sqrt{2} by 360.

$$x=\sqrt{2}+2$$

Now solve the equation x=\frac{720±360\sqrt{2}}{360} when ± is minus. Subtract 360\sqrt{2} from 720.

$$x=\frac{720-360\sqrt{2}}{360}$$

Divide 720-360\sqrt{2} by 360.

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

The equation is now solved.

$$x=\sqrt{2}+2 x=2-\sqrt{2}$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$180\left(x-2\right)x-180\left(x-2\right)=180x$$

Use the distributive property to multiply 180 by x-2.

$$\left(180x-360\right)x-180\left(x-2\right)=180x$$

Use the distributive property to multiply 180x-360 by x.

$$180x^{2}-360x-180\left(x-2\right)=180x$$

Use the distributive property to multiply -180 by x-2.

$$180x^{2}-360x-180x+360=180x$$

Combine -360x and -180x to get -540x.

$$180x^{2}-540x+360=180x$$

Subtract 180x from both sides.

$$180x^{2}-540x+360-180x=0$$

Combine -540x and -180x to get -720x.

$$180x^{2}-720x+360=0$$

Subtract 360 from both sides. Anything subtracted from zero gives its negation.

$$180x^{2}-720x=-360$$

Divide both sides by 180.

$$\frac{180x^{2}-720x}{180}=-\frac{360}{180}$$

Dividing by 180 undoes the multiplication by 180.

$$x^{2}+\left(-\frac{720}{180}\right)x=-\frac{360}{180}$$

Divide -720 by 180.

$$x^{2}-4x=-\frac{360}{180}$$

Divide -360 by 180.

$$x^{2}-4x=-2$$

Divide -4, the coefficient of the x term, by 2 to get -2. Then add the square of -2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -2.

$$x^{2}-4x+4=-2+4$$

Add -2 to 4.

$$x^{2}-4x+4=2$$

Factor x^{2}-4x+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}.

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

Take the square root of both sides of the equation.

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

Simplify.

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

Add 2 to both sides of the equation.

$$x=\sqrt{2}+2 x=2-\sqrt{2}$$