Solve 240/x+4=x/x-2

$$\frac{ 240 }{ x+4 } = \frac{ x }{ x-2 }$$

Solve for x

$$x=2\sqrt{3361}+118\approx 233.948264325$$

$$x=118-2\sqrt{3361}\approx 2.051735675$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -4,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x+4\right), the least common multiple of x+4,x-2.

$$\left(x-2\right)\times 240=\left(x+4\right)x$$

Use the distributive property to multiply x-2 by 240.

$$240x-480=\left(x+4\right)x$$

Use the distributive property to multiply x+4 by x.

$$240x-480=x^{2}+4x$$

Subtract x^{2} from both sides.

$$240x-480-x^{2}=4x$$

Subtract 4x from both sides.

$$240x-480-x^{2}-4x=0$$

Combine 240x and -4x to get 236x.

$$236x-480-x^{2}=0$$

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.

$$-x^{2}+236x-480=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 236 for b, and -480 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-236±\sqrt{236^{2}-4\left(-1\right)\left(-480\right)}}{2\left(-1\right)}$$

Square 236.

$$x=\frac{-236±\sqrt{55696-4\left(-1\right)\left(-480\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-236±\sqrt{55696+4\left(-480\right)}}{2\left(-1\right)}$$

Multiply 4 times -480.

$$x=\frac{-236±\sqrt{55696-1920}}{2\left(-1\right)}$$

Add 55696 to -1920.

$$x=\frac{-236±\sqrt{53776}}{2\left(-1\right)}$$

Take the square root of 53776.

$$x=\frac{-236±4\sqrt{3361}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-236±4\sqrt{3361}}{-2}$$

Now solve the equation x=\frac{-236±4\sqrt{3361}}{-2} when ± is plus. Add -236 to 4\sqrt{3361}.

$$x=\frac{4\sqrt{3361}-236}{-2}$$

Divide -236+4\sqrt{3361} by -2.

$$x=118-2\sqrt{3361}$$

Now solve the equation x=\frac{-236±4\sqrt{3361}}{-2} when ± is minus. Subtract 4\sqrt{3361} from -236.

$$x=\frac{-4\sqrt{3361}-236}{-2}$$

Divide -236-4\sqrt{3361} by -2.

$$x=2\sqrt{3361}+118$$

The equation is now solved.

$$x=118-2\sqrt{3361} x=2\sqrt{3361}+118$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -4,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x+4\right), the least common multiple of x+4,x-2.

$$\left(x-2\right)\times 240=\left(x+4\right)x$$

Use the distributive property to multiply x-2 by 240.

$$240x-480=\left(x+4\right)x$$

Use the distributive property to multiply x+4 by x.

$$240x-480=x^{2}+4x$$

Subtract x^{2} from both sides.

$$240x-480-x^{2}=4x$$

Subtract 4x from both sides.

$$240x-480-x^{2}-4x=0$$

Combine 240x and -4x to get 236x.

$$236x-480-x^{2}=0$$

Add 480 to both sides. Anything plus zero gives itself.

$$236x-x^{2}=480$$

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.

$$-x^{2}+236x=480$$

Divide both sides by -1.

$$\frac{-x^{2}+236x}{-1}=\frac{480}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{236}{-1}x=\frac{480}{-1}$$

Divide 236 by -1.

$$x^{2}-236x=\frac{480}{-1}$$

Divide 480 by -1.

$$x^{2}-236x=-480$$

Divide -236, the coefficient of the x term, by 2 to get -118. Then add the square of -118 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-236x+\left(-118\right)^{2}=-480+\left(-118\right)^{2}$$

Square -118.

$$x^{2}-236x+13924=-480+13924$$

Add -480 to 13924.

$$x^{2}-236x+13924=13444$$

Factor x^{2}-236x+13924. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-118\right)^{2}=13444$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-118\right)^{2}}=\sqrt{13444}$$

Simplify.

$$x-118=2\sqrt{3361} x-118=-2\sqrt{3361}$$

Add 118 to both sides of the equation.

$$x=2\sqrt{3361}+118 x=118-2\sqrt{3361}$$