Solve 200/x-1/2=200/x+15

$$\frac{ 200 }{ x } - \frac{ 1 }{ 2 } = \frac{ 200 }{ x+15 }$$

Solve for x

$$x = \frac{5 \sqrt{969} - 15}{2} \approx 70.321912081$$

$$x=\frac{-5\sqrt{969}-15}{2}\approx -85.321912081$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -15,0 since division by zero is not defined. Multiply both sides of the equation by 2x\left(x+15\right), the least common multiple of x,2,x+15.

$$\left(2x+30\right)\times 200+2x\left(x+15\right)\left(-\frac{1}{2}\right)=2x\times 200$$

Use the distributive property to multiply 2x+30 by 200.

$$400x+6000+2x\left(x+15\right)\left(-\frac{1}{2}\right)=2x\times 200$$

Multiply 2 and -\frac{1}{2} to get -1.

$$400x+6000-x\left(x+15\right)=2x\times 200$$

Use the distributive property to multiply -x by x+15.

$$400x+6000-x^{2}-15x=2x\times 200$$

Combine 400x and -15x to get 385x.

$$385x+6000-x^{2}=2x\times 200$$

Multiply 2 and 200 to get 400.

$$385x+6000-x^{2}=400x$$

Subtract 400x from both sides.

$$385x+6000-x^{2}-400x=0$$

Combine 385x and -400x to get -15x.

$$-15x+6000-x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-x^{2}-15x+6000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -15 for b, and 6000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\left(-1\right)\times 6000}}{2\left(-1\right)}$$

Square -15.

$$x=\frac{-\left(-15\right)±\sqrt{225-4\left(-1\right)\times 6000}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-15\right)±\sqrt{225+4\times 6000}}{2\left(-1\right)}$$

Multiply 4 times 6000.

$$x=\frac{-\left(-15\right)±\sqrt{225+24000}}{2\left(-1\right)}$$

Add 225 to 24000.

$$x=\frac{-\left(-15\right)±\sqrt{24225}}{2\left(-1\right)}$$

Take the square root of 24225.

$$x=\frac{-\left(-15\right)±5\sqrt{969}}{2\left(-1\right)}$$

The opposite of -15 is 15.

$$x=\frac{15±5\sqrt{969}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{15±5\sqrt{969}}{-2}$$

Now solve the equation x=\frac{15±5\sqrt{969}}{-2} when ± is plus. Add 15 to 5\sqrt{969}.

$$x=\frac{5\sqrt{969}+15}{-2}$$

Divide 15+5\sqrt{969} by -2.

$$x=\frac{-5\sqrt{969}-15}{2}$$

Now solve the equation x=\frac{15±5\sqrt{969}}{-2} when ± is minus. Subtract 5\sqrt{969} from 15.

$$x=\frac{15-5\sqrt{969}}{-2}$$

Divide 15-5\sqrt{969} by -2.

$$x=\frac{5\sqrt{969}-15}{2}$$

The equation is now solved.

$$x=\frac{-5\sqrt{969}-15}{2} x=\frac{5\sqrt{969}-15}{2}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -15,0 since division by zero is not defined. Multiply both sides of the equation by 2x\left(x+15\right), the least common multiple of x,2,x+15.

$$\left(2x+30\right)\times 200+2x\left(x+15\right)\left(-\frac{1}{2}\right)=2x\times 200$$

Use the distributive property to multiply 2x+30 by 200.

$$400x+6000+2x\left(x+15\right)\left(-\frac{1}{2}\right)=2x\times 200$$

Multiply 2 and -\frac{1}{2} to get -1.

$$400x+6000-x\left(x+15\right)=2x\times 200$$

Use the distributive property to multiply -x by x+15.

$$400x+6000-x^{2}-15x=2x\times 200$$

Combine 400x and -15x to get 385x.

$$385x+6000-x^{2}=2x\times 200$$

Multiply 2 and 200 to get 400.

$$385x+6000-x^{2}=400x$$

Subtract 400x from both sides.

$$385x+6000-x^{2}-400x=0$$

Combine 385x and -400x to get -15x.

$$-15x+6000-x^{2}=0$$

Subtract 6000 from both sides. Anything subtracted from zero gives its negation.

$$-15x-x^{2}=-6000$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-x^{2}-15x=-6000$$

Divide both sides by -1.

$$\frac{-x^{2}-15x}{-1}=-\frac{6000}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{15}{-1}\right)x=-\frac{6000}{-1}$$

Divide -15 by -1.

$$x^{2}+15x=-\frac{6000}{-1}$$

Divide -6000 by -1.

$$x^{2}+15x=6000$$

Divide 15, the coefficient of the x term, by 2 to get \frac{15}{2}. Then add the square of \frac{15}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+15x+\left(\frac{15}{2}\right)^{2}=6000+\left(\frac{15}{2}\right)^{2}$$

Square \frac{15}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+15x+\frac{225}{4}=6000+\frac{225}{4}$$

Add 6000 to \frac{225}{4}.

$$x^{2}+15x+\frac{225}{4}=\frac{24225}{4}$$

Factor x^{2}+15x+\frac{225}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{15}{2}\right)^{2}=\frac{24225}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{15}{2}\right)^{2}}=\sqrt{\frac{24225}{4}}$$

Simplify.

$$x+\frac{15}{2}=\frac{5\sqrt{969}}{2} x+\frac{15}{2}=-\frac{5\sqrt{969}}{2}$$

Subtract \frac{15}{2} from both sides of the equation.

$$x=\frac{5\sqrt{969}-15}{2} x=\frac{-5\sqrt{969}-15}{2}$$