Solve 1/x-8-1/x=2/15

$$\frac{ 1 }{ x-8 } - \frac{ 1 }{ x } = \frac{ 2 }{ 15 }$$

Solve for x

$$x=2\sqrt{19}+4\approx 12.717797887$$

$$x=4-2\sqrt{19}\approx -4.717797887$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,8 since division by zero is not defined. Multiply both sides of the equation by 15x\left(x-8\right), the least common multiple of x-8,x,15.

$$15x-\left(15x-120\right)=2x\left(x-8\right)$$

To find the opposite of 15x-120, find the opposite of each term.

$$15x-15x+120=2x\left(x-8\right)$$

Combine 15x and -15x to get 0.

$$120=2x\left(x-8\right)$$

Use the distributive property to multiply 2x by x-8.

$$120=2x^{2}-16x$$

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

$$2x^{2}-16x=120$$

Subtract 120 from both sides.

$$2x^{2}-16x-120=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -16 for b, and -120 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 2\left(-120\right)}}{2\times 2}$$

Square -16.

$$x=\frac{-\left(-16\right)±\sqrt{256-4\times 2\left(-120\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-16\right)±\sqrt{256-8\left(-120\right)}}{2\times 2}$$

Multiply -8 times -120.

$$x=\frac{-\left(-16\right)±\sqrt{256+960}}{2\times 2}$$

Add 256 to 960.

$$x=\frac{-\left(-16\right)±\sqrt{1216}}{2\times 2}$$

Take the square root of 1216.

$$x=\frac{-\left(-16\right)±8\sqrt{19}}{2\times 2}$$

The opposite of -16 is 16.

$$x=\frac{16±8\sqrt{19}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{16±8\sqrt{19}}{4}$$

Now solve the equation x=\frac{16±8\sqrt{19}}{4} when ± is plus. Add 16 to 8\sqrt{19}.

$$x=\frac{8\sqrt{19}+16}{4}$$

Divide 16+8\sqrt{19} by 4.

$$x=2\sqrt{19}+4$$

Now solve the equation x=\frac{16±8\sqrt{19}}{4} when ± is minus. Subtract 8\sqrt{19} from 16.

$$x=\frac{16-8\sqrt{19}}{4}$$

Divide 16-8\sqrt{19} by 4.

$$x=4-2\sqrt{19}$$

The equation is now solved.

$$x=2\sqrt{19}+4 x=4-2\sqrt{19}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,8 since division by zero is not defined. Multiply both sides of the equation by 15x\left(x-8\right), the least common multiple of x-8,x,15.

$$15x-\left(15x-120\right)=2x\left(x-8\right)$$

To find the opposite of 15x-120, find the opposite of each term.

$$15x-15x+120=2x\left(x-8\right)$$

Combine 15x and -15x to get 0.

$$120=2x\left(x-8\right)$$

Use the distributive property to multiply 2x by x-8.

$$120=2x^{2}-16x$$

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

$$2x^{2}-16x=120$$

Divide both sides by 2.

$$\frac{2x^{2}-16x}{2}=\frac{120}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{16}{2}\right)x=\frac{120}{2}$$

Divide -16 by 2.

$$x^{2}-8x=\frac{120}{2}$$

Divide 120 by 2.

$$x^{2}-8x=60$$

Divide -8, the coefficient of the x term, by 2 to get -4. Then add the square of -4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-8x+\left(-4\right)^{2}=60+\left(-4\right)^{2}$$

Square -4.

$$x^{2}-8x+16=60+16$$

Add 60 to 16.

$$x^{2}-8x+16=76$$

Factor x^{2}-8x+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-4\right)^{2}=76$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-4\right)^{2}}=\sqrt{76}$$

Simplify.

$$x-4=2\sqrt{19} x-4=-2\sqrt{19}$$

Add 4 to both sides of the equation.

$$x=2\sqrt{19}+4 x=4-2\sqrt{19}$$