Solve 360/x-2-360/x=2

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

Solve for x

$$x=-18$$

$$x=20$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,2 since division by zero is not defined. Multiply both sides of the equation by x\left(x-2\right), the least common multiple of x-2,x.

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

Use the distributive property to multiply x-2 by 360.

$$x\times 360-\left(360x-720\right)=2x\left(x-2\right)$$

To find the opposite of 360x-720, find the opposite of each term.

$$x\times 360-360x+720=2x\left(x-2\right)$$

Combine x\times 360 and -360x to get 0.

$$720=2x\left(x-2\right)$$

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

$$720=2x^{2}-4x$$

Swap sides so that all variable terms are on the left hand side.

$$2x^{2}-4x=720$$

Subtract 720 from both sides.

$$2x^{2}-4x-720=0$$

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

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

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 2\left(-720\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-4\right)±\sqrt{16-8\left(-720\right)}}{2\times 2}$$

Multiply -8 times -720.

$$x=\frac{-\left(-4\right)±\sqrt{16+5760}}{2\times 2}$$

Add 16 to 5760.

$$x=\frac{-\left(-4\right)±\sqrt{5776}}{2\times 2}$$

Take the square root of 5776.

$$x=\frac{-\left(-4\right)±76}{2\times 2}$$

The opposite of -4 is 4.

$$x=\frac{4±76}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{4±76}{4}$$

Now solve the equation x=\frac{4±76}{4} when ± is plus. Add 4 to 76.

$$x=\frac{80}{4}$$

Divide 80 by 4.

$$x=20$$

Now solve the equation x=\frac{4±76}{4} when ± is minus. Subtract 76 from 4.

$$x=-\frac{72}{4}$$

Divide -72 by 4.

$$x=-18$$

The equation is now solved.

$$x=20 x=-18$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,2 since division by zero is not defined. Multiply both sides of the equation by x\left(x-2\right), the least common multiple of x-2,x.

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

Use the distributive property to multiply x-2 by 360.

$$x\times 360-\left(360x-720\right)=2x\left(x-2\right)$$

To find the opposite of 360x-720, find the opposite of each term.

$$x\times 360-360x+720=2x\left(x-2\right)$$

Combine x\times 360 and -360x to get 0.

$$720=2x\left(x-2\right)$$

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

$$720=2x^{2}-4x$$

Swap sides so that all variable terms are on the left hand side.

$$2x^{2}-4x=720$$

Divide both sides by 2.

$$\frac{2x^{2}-4x}{2}=\frac{720}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{4}{2}\right)x=\frac{720}{2}$$

Divide -4 by 2.

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

Divide 720 by 2.

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

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.

$$x^{2}-2x+1=360+1$$

Add 360 to 1.

$$x^{2}-2x+1=361$$

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}.

$$\left(x-1\right)^{2}=361$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{361}$$

Simplify.

$$x-1=19 x-1=-19$$

Add 1 to both sides of the equation.

$$x=20 x=-18$$