Solve x/x^2-4+1/x^2-2x=11/6x^2+12x

$$\frac { x } { x ^ { 2 } - 4 } + \frac { 1 } { x ^ { 2 } - 2 x } = \frac { 11 } { 6 x ^ { 2 } + 12 x }$$

Solve for x (complex solution)

$$x=\frac{5+\sqrt{791}i}{12}\approx 0.416666667+2.343726852i$$

$$x=\frac{-\sqrt{791}i+5}{12}\approx 0.416666667-2.343726852i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -2,0,2 since division by zero is not defined. Multiply both sides of the equation by 6x\left(x-2\right)\left(x+2\right), the least common multiple of x^{2}-4,x^{2}-2x,6x^{2}+12x.

$$6xx+6x+12=\left(x-2\right)\times 11$$

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

$$6x^{2}+6x+12=\left(x-2\right)\times 11$$

Use the distributive property to multiply x-2 by 11.

$$6x^{2}+6x+12=11x-22$$

Subtract 11x from both sides.

$$6x^{2}+6x+12-11x=-22$$

Combine 6x and -11x to get -5x.

$$6x^{2}-5x+12=-22$$

Add 22 to both sides.

$$6x^{2}-5x+12+22=0$$

Add 12 and 22 to get 34.

$$6x^{2}-5x+34=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, -5 for b, and 34 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 6\times 34}}{2\times 6}$$

Square -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\times 6\times 34}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-\left(-5\right)±\sqrt{25-24\times 34}}{2\times 6}$$

Multiply -24 times 34.

$$x=\frac{-\left(-5\right)±\sqrt{25-816}}{2\times 6}$$

Add 25 to -816.

$$x=\frac{-\left(-5\right)±\sqrt{-791}}{2\times 6}$$

Take the square root of -791.

$$x=\frac{-\left(-5\right)±\sqrt{791}i}{2\times 6}$$

The opposite of -5 is 5.

$$x=\frac{5±\sqrt{791}i}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{5±\sqrt{791}i}{12}$$

Now solve the equation x=\frac{5±\sqrt{791}i}{12} when ± is plus. Add 5 to i\sqrt{791}.

$$x=\frac{5+\sqrt{791}i}{12}$$

Now solve the equation x=\frac{5±\sqrt{791}i}{12} when ± is minus. Subtract i\sqrt{791} from 5.

$$x=\frac{-\sqrt{791}i+5}{12}$$

The equation is now solved.

$$x=\frac{5+\sqrt{791}i}{12} x=\frac{-\sqrt{791}i+5}{12}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -2,0,2 since division by zero is not defined. Multiply both sides of the equation by 6x\left(x-2\right)\left(x+2\right), the least common multiple of x^{2}-4,x^{2}-2x,6x^{2}+12x.

$$6xx+6x+12=\left(x-2\right)\times 11$$

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

$$6x^{2}+6x+12=\left(x-2\right)\times 11$$

Use the distributive property to multiply x-2 by 11.

$$6x^{2}+6x+12=11x-22$$

Subtract 11x from both sides.

$$6x^{2}+6x+12-11x=-22$$

Combine 6x and -11x to get -5x.

$$6x^{2}-5x+12=-22$$

Subtract 12 from both sides.

$$6x^{2}-5x=-22-12$$

Subtract 12 from -22 to get -34.

$$6x^{2}-5x=-34$$

Divide both sides by 6.

$$\frac{6x^{2}-5x}{6}=-\frac{34}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}-\frac{5}{6}x=-\frac{34}{6}$$

Reduce the fraction \frac{-34}{6} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{5}{6}x=-\frac{17}{3}$$

Divide -\frac{5}{6}, the coefficient of the x term, by 2 to get -\frac{5}{12}. Then add the square of -\frac{5}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{5}{6}x+\left(-\frac{5}{12}\right)^{2}=-\frac{17}{3}+\left(-\frac{5}{12}\right)^{2}$$

Square -\frac{5}{12} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{5}{6}x+\frac{25}{144}=-\frac{17}{3}+\frac{25}{144}$$

Add -\frac{17}{3} to \frac{25}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{5}{6}x+\frac{25}{144}=-\frac{791}{144}$$

Factor x^{2}-\frac{5}{6}x+\frac{25}{144}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{5}{12}\right)^{2}=-\frac{791}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{12}\right)^{2}}=\sqrt{-\frac{791}{144}}$$

Simplify.

$$x-\frac{5}{12}=\frac{\sqrt{791}i}{12} x-\frac{5}{12}=-\frac{\sqrt{791}i}{12}$$

Add \frac{5}{12} to both sides of the equation.

$$x=\frac{5+\sqrt{791}i}{12} x=\frac{-\sqrt{791}i+5}{12}$$