Solve x+3/x-3+x-6/x+6=11

$$\frac { x + 3 } { x - 3 } + \frac { x - 6 } { x + 6 } = 11$$

Solve for x

$$x = \frac{\sqrt{1057} - 11}{6} \approx 3.585256069$$

$$x=\frac{-\sqrt{1057}-11}{6}\approx -7.251922736$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -6,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x+6\right), the least common multiple of x-3,x+6.

$$\left(x+6\right)\left(x+3\right)+\left(x-3\right)\left(x-6\right)=11\left(x-3\right)\left(x+6\right)$$

Use the distributive property to multiply x+6 by x+3 and combine like terms.

$$x^{2}+9x+18+\left(x-3\right)\left(x-6\right)=11\left(x-3\right)\left(x+6\right)$$

Use the distributive property to multiply x-3 by x-6 and combine like terms.

$$x^{2}+9x+18+x^{2}-9x+18=11\left(x-3\right)\left(x+6\right)$$

Combine x^{2} and x^{2} to get 2x^{2}.

$$2x^{2}+9x+18-9x+18=11\left(x-3\right)\left(x+6\right)$$

Combine 9x and -9x to get 0.

$$2x^{2}+18+18=11\left(x-3\right)\left(x+6\right)$$

Add 18 and 18 to get 36.

$$2x^{2}+36=11\left(x-3\right)\left(x+6\right)$$

Use the distributive property to multiply 11 by x-3.

$$2x^{2}+36=\left(11x-33\right)\left(x+6\right)$$

Use the distributive property to multiply 11x-33 by x+6 and combine like terms.

$$2x^{2}+36=11x^{2}+33x-198$$

Subtract 11x^{2} from both sides.

$$2x^{2}+36-11x^{2}=33x-198$$

Combine 2x^{2} and -11x^{2} to get -9x^{2}.

$$-9x^{2}+36=33x-198$$

Subtract 33x from both sides.

$$-9x^{2}+36-33x=-198$$

Add 198 to both sides.

$$-9x^{2}+36-33x+198=0$$

Add 36 and 198 to get 234.

$$-9x^{2}+234-33x=0$$

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.

$$-9x^{2}-33x+234=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -9 for a, -33 for b, and 234 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-33\right)±\sqrt{\left(-33\right)^{2}-4\left(-9\right)\times 234}}{2\left(-9\right)}$$

Square -33.

$$x=\frac{-\left(-33\right)±\sqrt{1089-4\left(-9\right)\times 234}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$x=\frac{-\left(-33\right)±\sqrt{1089+36\times 234}}{2\left(-9\right)}$$

Multiply 36 times 234.

$$x=\frac{-\left(-33\right)±\sqrt{1089+8424}}{2\left(-9\right)}$$

Add 1089 to 8424.

$$x=\frac{-\left(-33\right)±\sqrt{9513}}{2\left(-9\right)}$$

Take the square root of 9513.

$$x=\frac{-\left(-33\right)±3\sqrt{1057}}{2\left(-9\right)}$$

The opposite of -33 is 33.

$$x=\frac{33±3\sqrt{1057}}{2\left(-9\right)}$$

Multiply 2 times -9.

$$x=\frac{33±3\sqrt{1057}}{-18}$$

Now solve the equation x=\frac{33±3\sqrt{1057}}{-18} when ± is plus. Add 33 to 3\sqrt{1057}.

$$x=\frac{3\sqrt{1057}+33}{-18}$$

Divide 33+3\sqrt{1057} by -18.

$$x=\frac{-\sqrt{1057}-11}{6}$$

Now solve the equation x=\frac{33±3\sqrt{1057}}{-18} when ± is minus. Subtract 3\sqrt{1057} from 33.

$$x=\frac{33-3\sqrt{1057}}{-18}$$

Divide 33-3\sqrt{1057} by -18.

$$x=\frac{\sqrt{1057}-11}{6}$$

The equation is now solved.

$$x=\frac{-\sqrt{1057}-11}{6} x=\frac{\sqrt{1057}-11}{6}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -6,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x+6\right), the least common multiple of x-3,x+6.

$$\left(x+6\right)\left(x+3\right)+\left(x-3\right)\left(x-6\right)=11\left(x-3\right)\left(x+6\right)$$

Use the distributive property to multiply x+6 by x+3 and combine like terms.

$$x^{2}+9x+18+\left(x-3\right)\left(x-6\right)=11\left(x-3\right)\left(x+6\right)$$

Use the distributive property to multiply x-3 by x-6 and combine like terms.

$$x^{2}+9x+18+x^{2}-9x+18=11\left(x-3\right)\left(x+6\right)$$

Combine x^{2} and x^{2} to get 2x^{2}.

$$2x^{2}+9x+18-9x+18=11\left(x-3\right)\left(x+6\right)$$

Combine 9x and -9x to get 0.

$$2x^{2}+18+18=11\left(x-3\right)\left(x+6\right)$$

Add 18 and 18 to get 36.

$$2x^{2}+36=11\left(x-3\right)\left(x+6\right)$$

Use the distributive property to multiply 11 by x-3.

$$2x^{2}+36=\left(11x-33\right)\left(x+6\right)$$

Use the distributive property to multiply 11x-33 by x+6 and combine like terms.

$$2x^{2}+36=11x^{2}+33x-198$$

Subtract 11x^{2} from both sides.

$$2x^{2}+36-11x^{2}=33x-198$$

Combine 2x^{2} and -11x^{2} to get -9x^{2}.

$$-9x^{2}+36=33x-198$$

Subtract 33x from both sides.

$$-9x^{2}+36-33x=-198$$

Subtract 36 from both sides.

$$-9x^{2}-33x=-198-36$$

Subtract 36 from -198 to get -234.

$$-9x^{2}-33x=-234$$

Divide both sides by -9.

$$\frac{-9x^{2}-33x}{-9}=-\frac{234}{-9}$$

Dividing by -9 undoes the multiplication by -9.

$$x^{2}+\left(-\frac{33}{-9}\right)x=-\frac{234}{-9}$$

Reduce the fraction \frac{-33}{-9} to lowest terms by extracting and canceling out 3.

$$x^{2}+\frac{11}{3}x=-\frac{234}{-9}$$

Divide -234 by -9.

$$x^{2}+\frac{11}{3}x=26$$

Divide \frac{11}{3}, the coefficient of the x term, by 2 to get \frac{11}{6}. Then add the square of \frac{11}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{11}{3}x+\left(\frac{11}{6}\right)^{2}=26+\left(\frac{11}{6}\right)^{2}$$

Square \frac{11}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{11}{3}x+\frac{121}{36}=26+\frac{121}{36}$$

Add 26 to \frac{121}{36}.

$$x^{2}+\frac{11}{3}x+\frac{121}{36}=\frac{1057}{36}$$

Factor x^{2}+\frac{11}{3}x+\frac{121}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{11}{6}\right)^{2}=\frac{1057}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{11}{6}\right)^{2}}=\sqrt{\frac{1057}{36}}$$

Simplify.

$$x+\frac{11}{6}=\frac{\sqrt{1057}}{6} x+\frac{11}{6}=-\frac{\sqrt{1057}}{6}$$

Subtract \frac{11}{6} from both sides of the equation.

$$x=\frac{\sqrt{1057}-11}{6} x=\frac{-\sqrt{1057}-11}{6}$$