Solve x=15/x+5

$$x= \frac{ 15 }{ x+5 }$$

Solve for x

$$x = \frac{\sqrt{85} - 5}{2} \approx 2.109772229$$

$$x=\frac{-\sqrt{85}-5}{2}\approx -7.109772229$$

Steps Using the Quadratic Formula

Subtract \frac{15}{x+5} from both sides.

$$x-\frac{15}{x+5}=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{x+5}{x+5}.

$$\frac{x\left(x+5\right)}{x+5}-\frac{15}{x+5}=0$$

Since \frac{x\left(x+5\right)}{x+5} and \frac{15}{x+5} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x\left(x+5\right)-15}{x+5}=0$$

Do the multiplications in x\left(x+5\right)-15.

$$\frac{x^{2}+5x-15}{x+5}=0$$

Variable x cannot be equal to -5 since division by zero is not defined. Multiply both sides of the equation by x+5.

$$x^{2}+5x-15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 5 for b, and -15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\left(-15\right)}}{2}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\left(-15\right)}}{2}$$

Multiply -4 times -15.

$$x=\frac{-5±\sqrt{25+60}}{2}$$

Add 25 to 60.

$$x=\frac{-5±\sqrt{85}}{2}$$

Now solve the equation x=\frac{-5±\sqrt{85}}{2} when ± is plus. Add -5 to \sqrt{85}.

$$x=\frac{\sqrt{85}-5}{2}$$

Now solve the equation x=\frac{-5±\sqrt{85}}{2} when ± is minus. Subtract \sqrt{85} from -5.

$$x=\frac{-\sqrt{85}-5}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{85}-5}{2} x=\frac{-\sqrt{85}-5}{2}$$

Steps for Completing the Square

Subtract \frac{15}{x+5} from both sides.

$$x-\frac{15}{x+5}=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{x+5}{x+5}.

$$\frac{x\left(x+5\right)}{x+5}-\frac{15}{x+5}=0$$

Since \frac{x\left(x+5\right)}{x+5} and \frac{15}{x+5} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x\left(x+5\right)-15}{x+5}=0$$

Do the multiplications in x\left(x+5\right)-15.

$$\frac{x^{2}+5x-15}{x+5}=0$$

Variable x cannot be equal to -5 since division by zero is not defined. Multiply both sides of the equation by x+5.

$$x^{2}+5x-15=0$$

Add 15 to both sides. Anything plus zero gives itself.

$$x^{2}+5x=15$$

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

$$x^{2}+5x+\left(\frac{5}{2}\right)^{2}=15+\left(\frac{5}{2}\right)^{2}$$

Square \frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+5x+\frac{25}{4}=15+\frac{25}{4}$$

Add 15 to \frac{25}{4}.

$$x^{2}+5x+\frac{25}{4}=\frac{85}{4}$$

Factor x^{2}+5x+\frac{25}{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}.

$$\left(x+\frac{5}{2}\right)^{2}=\frac{85}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{5}{2}\right)^{2}}=\sqrt{\frac{85}{4}}$$

Simplify.

$$x+\frac{5}{2}=\frac{\sqrt{85}}{2} x+\frac{5}{2}=-\frac{\sqrt{85}}{2}$$

Subtract \frac{5}{2} from both sides of the equation.

$$x=\frac{\sqrt{85}-5}{2} x=\frac{-\sqrt{85}-5}{2}$$