Solve 240/x-2-240/x=4

$$\frac{ 240 }{ x-2 } - \frac{ 240 }{ x } =4$$

Solve for x

$$x=-10$$

$$x=12$$

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.

$$x\times 240-\left(x-2\right)\times 240=4x\left(x-2\right)$$

Use the distributive property to multiply x-2 by 240.

$$x\times 240-\left(240x-480\right)=4x\left(x-2\right)$$

To find the opposite of 240x-480, find the opposite of each term.

$$x\times 240-240x+480=4x\left(x-2\right)$$

Combine x\times 240 and -240x to get 0.

$$480=4x\left(x-2\right)$$

Use the distributive property to multiply 4x by x-2.

$$480=4x^{2}-8x$$

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

$$4x^{2}-8x=480$$

Subtract 480 from both sides.

$$4x^{2}-8x-480=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -8 for b, and -480 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 4\left(-480\right)}}{2\times 4}$$

Square -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\times 4\left(-480\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-8\right)±\sqrt{64-16\left(-480\right)}}{2\times 4}$$

Multiply -16 times -480.

$$x=\frac{-\left(-8\right)±\sqrt{64+7680}}{2\times 4}$$

Add 64 to 7680.

$$x=\frac{-\left(-8\right)±\sqrt{7744}}{2\times 4}$$

Take the square root of 7744.

$$x=\frac{-\left(-8\right)±88}{2\times 4}$$

The opposite of -8 is 8.

$$x=\frac{8±88}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{8±88}{8}$$

Now solve the equation x=\frac{8±88}{8} when ± is plus. Add 8 to 88.

$$x=\frac{96}{8}$$

Divide 96 by 8.

$$x=12$$

Now solve the equation x=\frac{8±88}{8} when ± is minus. Subtract 88 from 8.

$$x=-\frac{80}{8}$$

Divide -80 by 8.

$$x=-10$$

The equation is now solved.

$$x=12 x=-10$$

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.

$$x\times 240-\left(x-2\right)\times 240=4x\left(x-2\right)$$

Use the distributive property to multiply x-2 by 240.

$$x\times 240-\left(240x-480\right)=4x\left(x-2\right)$$

To find the opposite of 240x-480, find the opposite of each term.

$$x\times 240-240x+480=4x\left(x-2\right)$$

Combine x\times 240 and -240x to get 0.

$$480=4x\left(x-2\right)$$

Use the distributive property to multiply 4x by x-2.

$$480=4x^{2}-8x$$

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

$$4x^{2}-8x=480$$

Divide both sides by 4.

$$\frac{4x^{2}-8x}{4}=\frac{480}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\left(-\frac{8}{4}\right)x=\frac{480}{4}$$

Divide -8 by 4.

$$x^{2}-2x=\frac{480}{4}$$

Divide 480 by 4.

$$x^{2}-2x=120$$

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.

$$x^{2}-2x+1=120+1$$

Add 120 to 1.

$$x^{2}-2x+1=121$$

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}.

$$\left(x-1\right)^{2}=121$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{121}$$

Simplify.

$$x-1=11 x-1=-11$$

Add 1 to both sides of the equation.

$$x=12 x=-10$$