Solve 9/x+12+9/x-12=2

$$\frac { 9 } { x + 12 } + \frac { 9 } { x - 12 } = 2$$

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.

$$\left(x-12\right)\times 9+\left(x+12\right)\times 9=2\left(x-12\right)\left(x+12\right)$$

Use the distributive property to multiply x-12 by 9.

$$9x-108+\left(x+12\right)\times 9=2\left(x-12\right)\left(x+12\right)$$

Use the distributive property to multiply x+12 by 9.

$$9x-108+9x+108=2\left(x-12\right)\left(x+12\right)$$

Combine 9x and 9x to get 18x.

$$18x-108+108=2\left(x-12\right)\left(x+12\right)$$

Add -108 and 108 to get 0.

$$18x=2\left(x-12\right)\left(x+12\right)$$

Use the distributive property to multiply 2 by x-12.

$$18x=\left(2x-24\right)\left(x+12\right)$$

Use the distributive property to multiply 2x-24 by x+12 and combine like terms.

$$18x=2x^{2}-288$$

Subtract 2x^{2} from both sides.

$$18x-2x^{2}=-288$$

Add 288 to both sides.

$$18x-2x^{2}+288=0$$

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.

$$-2x^{2}+18x+288=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 18 for b, and 288 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-18±\sqrt{18^{2}-4\left(-2\right)\times 288}}{2\left(-2\right)}$$

Square 18.

$$x=\frac{-18±\sqrt{324-4\left(-2\right)\times 288}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-18±\sqrt{324+8\times 288}}{2\left(-2\right)}$$

Multiply 8 times 288.

$$x=\frac{-18±\sqrt{324+2304}}{2\left(-2\right)}$$

Add 324 to 2304.

$$x=\frac{-18±\sqrt{2628}}{2\left(-2\right)}$$

Take the square root of 2628.

$$x=\frac{-18±6\sqrt{73}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-18±6\sqrt{73}}{-4}$$

Now solve the equation x=\frac{-18±6\sqrt{73}}{-4} when ± is plus. Add -18 to 6\sqrt{73}.

$$x=\frac{6\sqrt{73}-18}{-4}$$

Divide -18+6\sqrt{73} by -4.

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

Now solve the equation x=\frac{-18±6\sqrt{73}}{-4} when ± is minus. Subtract 6\sqrt{73} from -18.

$$x=\frac{-6\sqrt{73}-18}{-4}$$

Divide -18-6\sqrt{73} by -4.

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

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.

$$\left(x-12\right)\times 9+\left(x+12\right)\times 9=2\left(x-12\right)\left(x+12\right)$$

Use the distributive property to multiply x-12 by 9.

$$9x-108+\left(x+12\right)\times 9=2\left(x-12\right)\left(x+12\right)$$

Use the distributive property to multiply x+12 by 9.

$$9x-108+9x+108=2\left(x-12\right)\left(x+12\right)$$

Combine 9x and 9x to get 18x.

$$18x-108+108=2\left(x-12\right)\left(x+12\right)$$

Add -108 and 108 to get 0.

$$18x=2\left(x-12\right)\left(x+12\right)$$

Use the distributive property to multiply 2 by x-12.

$$18x=\left(2x-24\right)\left(x+12\right)$$

Use the distributive property to multiply 2x-24 by x+12 and combine like terms.

$$18x=2x^{2}-288$$

Subtract 2x^{2} from both sides.

$$18x-2x^{2}=-288$$

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.

$$-2x^{2}+18x=-288$$

Divide both sides by -2.

$$\frac{-2x^{2}+18x}{-2}=-\frac{288}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{18}{-2}x=-\frac{288}{-2}$$

Divide 18 by -2.

$$x^{2}-9x=-\frac{288}{-2}$$

Divide -288 by -2.

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

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.

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