Solve 36/x-6-36/x=1/5

$$\frac { 36 } { x - 6 } - \frac { 36 } { x } = \frac { 1 } { 5 }$$

Solve for x

$$x=-30$$

$$x=36$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,6 since division by zero is not defined. Multiply both sides of the equation by 5x\left(x-6\right), the least common multiple of x-6,x,5.

$$5x\times 36-\left(5x-30\right)\times 36=x\left(x-6\right)$$

Multiply 5 and 36 to get 180.

$$180x-\left(5x-30\right)\times 36=x\left(x-6\right)$$

Use the distributive property to multiply 5x-30 by 36.

$$180x-\left(180x-1080\right)=x\left(x-6\right)$$

To find the opposite of 180x-1080, find the opposite of each term.

$$180x-180x+1080=x\left(x-6\right)$$

Combine 180x and -180x to get 0.

$$1080=x\left(x-6\right)$$

Use the distributive property to multiply x by x-6.

$$1080=x^{2}-6x$$

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

$$x^{2}-6x=1080$$

Subtract 1080 from both sides.

$$x^{2}-6x-1080=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -6 for b, and -1080 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-1080\right)}}{2}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\left(-1080\right)}}{2}$$

Multiply -4 times -1080.

$$x=\frac{-\left(-6\right)±\sqrt{36+4320}}{2}$$

Add 36 to 4320.

$$x=\frac{-\left(-6\right)±\sqrt{4356}}{2}$$

Take the square root of 4356.

$$x=\frac{-\left(-6\right)±66}{2}$$

The opposite of -6 is 6.

$$x=\frac{6±66}{2}$$

Now solve the equation x=\frac{6±66}{2} when ± is plus. Add 6 to 66.

$$x=\frac{72}{2}$$

Divide 72 by 2.

$$x=36$$

Now solve the equation x=\frac{6±66}{2} when ± is minus. Subtract 66 from 6.

$$x=-\frac{60}{2}$$

Divide -60 by 2.

$$x=-30$$

The equation is now solved.

$$x=36 x=-30$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,6 since division by zero is not defined. Multiply both sides of the equation by 5x\left(x-6\right), the least common multiple of x-6,x,5.

$$5x\times 36-\left(5x-30\right)\times 36=x\left(x-6\right)$$

Multiply 5 and 36 to get 180.

$$180x-\left(5x-30\right)\times 36=x\left(x-6\right)$$

Use the distributive property to multiply 5x-30 by 36.

$$180x-\left(180x-1080\right)=x\left(x-6\right)$$

To find the opposite of 180x-1080, find the opposite of each term.

$$180x-180x+1080=x\left(x-6\right)$$

Combine 180x and -180x to get 0.

$$1080=x\left(x-6\right)$$

Use the distributive property to multiply x by x-6.

$$1080=x^{2}-6x$$

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

$$x^{2}-6x=1080$$

Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-6x+\left(-3\right)^{2}=1080+\left(-3\right)^{2}$$

Square -3.

$$x^{2}-6x+9=1080+9$$

Add 1080 to 9.

$$x^{2}-6x+9=1089$$

Factor x^{2}-6x+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-3\right)^{2}=1089$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{1089}$$

Simplify.

$$x-3=33 x-3=-33$$

Add 3 to both sides of the equation.

$$x=36 x=-30$$