Solve 36/x+12+36/x-12=8

$$\frac{ 36 }{ x+12 } + \frac{ 36 }{ x-12 } = 8$$

Solve for x

$$x = \frac{3 \sqrt{73} + 9}{2} \approx 17.316005618$$

$$x=\frac{9-3\sqrt{73}}{2}\approx -8.316005618$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -12,12 since division by zero is not defined. Multiply both sides of the equation by \left(x-12\right)\left(x+12\right), the least common multiple of x+12,x-12.

$$\left(x-12\right)\times 36+\left(x+12\right)\times 36=8\left(x-12\right)\left(x+12\right)$$

Use the distributive property to multiply x-12 by 36.

$$36x-432+\left(x+12\right)\times 36=8\left(x-12\right)\left(x+12\right)$$

Use the distributive property to multiply x+12 by 36.

$$36x-432+36x+432=8\left(x-12\right)\left(x+12\right)$$

Combine 36x and 36x to get 72x.

$$72x-432+432=8\left(x-12\right)\left(x+12\right)$$

Add -432 and 432 to get 0.

$$72x=8\left(x-12\right)\left(x+12\right)$$

Use the distributive property to multiply 8 by x-12.

$$72x=\left(8x-96\right)\left(x+12\right)$$

Use the distributive property to multiply 8x-96 by x+12 and combine like terms.

$$72x=8x^{2}-1152$$

Subtract 8x^{2} from both sides.

$$72x-8x^{2}=-1152$$

Add 1152 to both sides.

$$72x-8x^{2}+1152=0$$

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.

$$-8x^{2}+72x+1152=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -8 for a, 72 for b, and 1152 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-72±\sqrt{72^{2}-4\left(-8\right)\times 1152}}{2\left(-8\right)}$$

Square 72.

$$x=\frac{-72±\sqrt{5184-4\left(-8\right)\times 1152}}{2\left(-8\right)}$$

Multiply -4 times -8.

$$x=\frac{-72±\sqrt{5184+32\times 1152}}{2\left(-8\right)}$$

Multiply 32 times 1152.

$$x=\frac{-72±\sqrt{5184+36864}}{2\left(-8\right)}$$

Add 5184 to 36864.

$$x=\frac{-72±\sqrt{42048}}{2\left(-8\right)}$$

Take the square root of 42048.

$$x=\frac{-72±24\sqrt{73}}{2\left(-8\right)}$$

Multiply 2 times -8.

$$x=\frac{-72±24\sqrt{73}}{-16}$$

Now solve the equation x=\frac{-72±24\sqrt{73}}{-16} when ± is plus. Add -72 to 24\sqrt{73}.

$$x=\frac{24\sqrt{73}-72}{-16}$$

Divide -72+24\sqrt{73} by -16.

$$x=\frac{9-3\sqrt{73}}{2}$$

Now solve the equation x=\frac{-72±24\sqrt{73}}{-16} when ± is minus. Subtract 24\sqrt{73} from -72.

$$x=\frac{-24\sqrt{73}-72}{-16}$$

Divide -72-24\sqrt{73} by -16.

$$x=\frac{3\sqrt{73}+9}{2}$$

The equation is now solved.

$$x=\frac{9-3\sqrt{73}}{2} x=\frac{3\sqrt{73}+9}{2}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -12,12 since division by zero is not defined. Multiply both sides of the equation by \left(x-12\right)\left(x+12\right), the least common multiple of x+12,x-12.

$$\left(x-12\right)\times 36+\left(x+12\right)\times 36=8\left(x-12\right)\left(x+12\right)$$

Use the distributive property to multiply x-12 by 36.

$$36x-432+\left(x+12\right)\times 36=8\left(x-12\right)\left(x+12\right)$$

Use the distributive property to multiply x+12 by 36.

$$36x-432+36x+432=8\left(x-12\right)\left(x+12\right)$$

Combine 36x and 36x to get 72x.

$$72x-432+432=8\left(x-12\right)\left(x+12\right)$$

Add -432 and 432 to get 0.

$$72x=8\left(x-12\right)\left(x+12\right)$$

Use the distributive property to multiply 8 by x-12.

$$72x=\left(8x-96\right)\left(x+12\right)$$

Use the distributive property to multiply 8x-96 by x+12 and combine like terms.

$$72x=8x^{2}-1152$$

Subtract 8x^{2} from both sides.

$$72x-8x^{2}=-1152$$

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.

$$-8x^{2}+72x=-1152$$

Divide both sides by -8.

$$\frac{-8x^{2}+72x}{-8}=-\frac{1152}{-8}$$

Dividing by -8 undoes the multiplication by -8.

$$x^{2}+\frac{72}{-8}x=-\frac{1152}{-8}$$

Divide 72 by -8.

$$x^{2}-9x=-\frac{1152}{-8}$$

Divide -1152 by -8.

$$x^{2}-9x=144$$

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.

$$x^{2}-9x+\left(-\frac{9}{2}\right)^{2}=144+\left(-\frac{9}{2}\right)^{2}$$

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

$$x^{2}-9x+\frac{81}{4}=144+\frac{81}{4}$$

Add 144 to \frac{81}{4}.

$$x^{2}-9x+\frac{81}{4}=\frac{657}{4}$$

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}.

$$\left(x-\frac{9}{2}\right)^{2}=\frac{657}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{2}\right)^{2}}=\sqrt{\frac{657}{4}}$$

Simplify.

$$x-\frac{9}{2}=\frac{3\sqrt{73}}{2} x-\frac{9}{2}=-\frac{3\sqrt{73}}{2}$$

Add \frac{9}{2} to both sides of the equation.

$$x=\frac{3\sqrt{73}+9}{2} x=\frac{9-3\sqrt{73}}{2}$$