Solve 240/x+14=x

$$\frac{ 240 }{ x+14 } =x$$

Solve for x

$$x=-24$$

$$x=10$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -14 since division by zero is not defined. Multiply both sides of the equation by x+14.

$$240=x\left(x+14\right)$$

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

$$240=x^{2}+14x$$

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

$$x^{2}+14x=240$$

Subtract 240 from both sides.

$$x^{2}+14x-240=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 14 for b, and -240 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-14±\sqrt{14^{2}-4\left(-240\right)}}{2}$$

Square 14.

$$x=\frac{-14±\sqrt{196-4\left(-240\right)}}{2}$$

Multiply -4 times -240.

$$x=\frac{-14±\sqrt{196+960}}{2}$$

Add 196 to 960.

$$x=\frac{-14±\sqrt{1156}}{2}$$

Take the square root of 1156.

$$x=\frac{-14±34}{2}$$

Now solve the equation x=\frac{-14±34}{2} when ± is plus. Add -14 to 34.

$$x=\frac{20}{2}$$

Divide 20 by 2.

$$x=10$$

Now solve the equation x=\frac{-14±34}{2} when ± is minus. Subtract 34 from -14.

$$x=-\frac{48}{2}$$

Divide -48 by 2.

$$x=-24$$

The equation is now solved.

$$x=10 x=-24$$

Steps for Completing the Square

Variable x cannot be equal to -14 since division by zero is not defined. Multiply both sides of the equation by x+14.

$$240=x\left(x+14\right)$$

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

$$240=x^{2}+14x$$

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

$$x^{2}+14x=240$$

Divide 14, the coefficient of the x term, by 2 to get 7. Then add the square of 7 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+14x+7^{2}=240+7^{2}$$

Square 7.

$$x^{2}+14x+49=240+49$$

Add 240 to 49.

$$x^{2}+14x+49=289$$

Factor x^{2}+14x+49. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+7\right)^{2}=289$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+7\right)^{2}}=\sqrt{289}$$

Simplify.

$$x+7=17 x+7=-17$$

Subtract 7 from both sides of the equation.

$$x=10 x=-24$$