Solve x+14/x+5-x=x+5/x

$$\frac { x + 14 } { x + 5 - x } = \frac { x + 5 } { x }$$

Solve for x

$$x = \frac{\sqrt{181} - 9}{2} \approx 2.226812024$$

$$x=\frac{-\sqrt{181}-9}{2}\approx -11.226812024$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 5x, the least common multiple of x+5-x,x.

$$x\left(x+14\right)=5\left(x+5\right)$$

Use the distributive property to multiply x by x+14.

$$x^{2}+14x=5\left(x+5\right)$$

Use the distributive property to multiply 5 by x+5.

$$x^{2}+14x=5x+25$$

Subtract 5x from both sides.

$$x^{2}+14x-5x=25$$

Combine 14x and -5x to get 9x.

$$x^{2}+9x=25$$

Subtract 25 from both sides.

$$x^{2}+9x-25=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 9 for b, and -25 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-9±\sqrt{9^{2}-4\left(-25\right)}}{2}$$

Square 9.

$$x=\frac{-9±\sqrt{81-4\left(-25\right)}}{2}$$

Multiply -4 times -25.

$$x=\frac{-9±\sqrt{81+100}}{2}$$

Add 81 to 100.

$$x=\frac{-9±\sqrt{181}}{2}$$

Now solve the equation x=\frac{-9±\sqrt{181}}{2} when ± is plus. Add -9 to \sqrt{181}.

$$x=\frac{\sqrt{181}-9}{2}$$

Now solve the equation x=\frac{-9±\sqrt{181}}{2} when ± is minus. Subtract \sqrt{181} from -9.

$$x=\frac{-\sqrt{181}-9}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{181}-9}{2} x=\frac{-\sqrt{181}-9}{2}$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 5x, the least common multiple of x+5-x,x.

$$x\left(x+14\right)=5\left(x+5\right)$$

Use the distributive property to multiply x by x+14.

$$x^{2}+14x=5\left(x+5\right)$$

Use the distributive property to multiply 5 by x+5.

$$x^{2}+14x=5x+25$$

Subtract 5x from both sides.

$$x^{2}+14x-5x=25$$

Combine 14x and -5x to get 9x.

$$x^{2}+9x=25$$

Divide 9, the coefficient of the x term, by 2 to get \frac{9}{2}. Then add the square of \frac{9}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+9x+\left(\frac{9}{2}\right)^{2}=25+\left(\frac{9}{2}\right)^{2}$$

Square \frac{9}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+9x+\frac{81}{4}=25+\frac{81}{4}$$

Add 25 to \frac{81}{4}.

$$x^{2}+9x+\frac{81}{4}=\frac{181}{4}$$

Factor x^{2}+9x+\frac{81}{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}.

$$\left(x+\frac{9}{2}\right)^{2}=\frac{181}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{9}{2}\right)^{2}}=\sqrt{\frac{181}{4}}$$

Simplify.

$$x+\frac{9}{2}=\frac{\sqrt{181}}{2} x+\frac{9}{2}=-\frac{\sqrt{181}}{2}$$

Subtract \frac{9}{2} from both sides of the equation.

$$x=\frac{\sqrt{181}-9}{2} x=\frac{-\sqrt{181}-9}{2}$$