Solve 1/x-10-1/x=8/75

$$\frac { 1 } { x - 10 } - \frac { 1 } { x } = \frac { 8 } { 75 }$$

Solve for x

$$x=\frac{5\sqrt{19}}{2}+5\approx 15.897247359$$

$$x=-\frac{5\sqrt{19}}{2}+5\approx -5.897247359$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,10 since division by zero is not defined. Multiply both sides of the equation by 75x\left(x-10\right), the least common multiple of x-10,x,75.

$$75x-\left(75x-750\right)=8x\left(x-10\right)$$

To find the opposite of 75x-750, find the opposite of each term.

$$75x-75x+750=8x\left(x-10\right)$$

Combine 75x and -75x to get 0.

$$750=8x\left(x-10\right)$$

Use the distributive property to multiply 8x by x-10.

$$750=8x^{2}-80x$$

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

$$8x^{2}-80x=750$$

Subtract 750 from both sides.

$$8x^{2}-80x-750=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 8 for a, -80 for b, and -750 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-80\right)±\sqrt{\left(-80\right)^{2}-4\times 8\left(-750\right)}}{2\times 8}$$

Square -80.

$$x=\frac{-\left(-80\right)±\sqrt{6400-4\times 8\left(-750\right)}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-\left(-80\right)±\sqrt{6400-32\left(-750\right)}}{2\times 8}$$

Multiply -32 times -750.

$$x=\frac{-\left(-80\right)±\sqrt{6400+24000}}{2\times 8}$$

Add 6400 to 24000.

$$x=\frac{-\left(-80\right)±\sqrt{30400}}{2\times 8}$$

Take the square root of 30400.

$$x=\frac{-\left(-80\right)±40\sqrt{19}}{2\times 8}$$

The opposite of -80 is 80.

$$x=\frac{80±40\sqrt{19}}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{80±40\sqrt{19}}{16}$$

Now solve the equation x=\frac{80±40\sqrt{19}}{16} when ± is plus. Add 80 to 40\sqrt{19}.

$$x=\frac{40\sqrt{19}+80}{16}$$

Divide 80+40\sqrt{19} by 16.

$$x=\frac{5\sqrt{19}}{2}+5$$

Now solve the equation x=\frac{80±40\sqrt{19}}{16} when ± is minus. Subtract 40\sqrt{19} from 80.

$$x=\frac{80-40\sqrt{19}}{16}$$

Divide 80-40\sqrt{19} by 16.

$$x=-\frac{5\sqrt{19}}{2}+5$$

The equation is now solved.

$$x=\frac{5\sqrt{19}}{2}+5 x=-\frac{5\sqrt{19}}{2}+5$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,10 since division by zero is not defined. Multiply both sides of the equation by 75x\left(x-10\right), the least common multiple of x-10,x,75.

$$75x-\left(75x-750\right)=8x\left(x-10\right)$$

To find the opposite of 75x-750, find the opposite of each term.

$$75x-75x+750=8x\left(x-10\right)$$

Combine 75x and -75x to get 0.

$$750=8x\left(x-10\right)$$

Use the distributive property to multiply 8x by x-10.

$$750=8x^{2}-80x$$

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

$$8x^{2}-80x=750$$

Divide both sides by 8.

$$\frac{8x^{2}-80x}{8}=\frac{750}{8}$$

Dividing by 8 undoes the multiplication by 8.

$$x^{2}+\left(-\frac{80}{8}\right)x=\frac{750}{8}$$

Divide -80 by 8.

$$x^{2}-10x=\frac{750}{8}$$

Reduce the fraction \frac{750}{8} to lowest terms by extracting and canceling out 2.

$$x^{2}-10x=\frac{375}{4}$$

Divide -10, the coefficient of the x term, by 2 to get -5. Then add the square of -5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-10x+\left(-5\right)^{2}=\frac{375}{4}+\left(-5\right)^{2}$$

Square -5.

$$x^{2}-10x+25=\frac{375}{4}+25$$

Add \frac{375}{4} to 25.

$$x^{2}-10x+25=\frac{475}{4}$$

Factor x^{2}-10x+25. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-5\right)^{2}=\frac{475}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-5\right)^{2}}=\sqrt{\frac{475}{4}}$$

Simplify.

$$x-5=\frac{5\sqrt{19}}{2} x-5=-\frac{5\sqrt{19}}{2}$$

Add 5 to both sides of the equation.

$$x=\frac{5\sqrt{19}}{2}+5 x=-\frac{5\sqrt{19}}{2}+5$$