Solve 1/3+1/x=1/2+x-1/6x

$$\frac { 1 } { 3 } + \frac { 1 } { x } = \frac { 1 } { 2 + x } - \frac { 1 } { 6 x }$$

Solve for x (complex solution)

$$x=\frac{-5+\sqrt{87}i}{4}\approx -1.25+2.331844763i$$

$$x=\frac{-\sqrt{87}i-5}{4}\approx -1.25-2.331844763i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -2,0 since division by zero is not defined. Multiply both sides of the equation by 6x\left(x+2\right), the least common multiple of 3,x,2+x,6x.

$$6x\left(x+2\right)\times \frac{1}{3}+6x+12=6x-\left(x+2\right)$$

Use the distributive property to multiply 6x by x+2.

$$\left(6x^{2}+12x\right)\times \frac{1}{3}+6x+12=6x-\left(x+2\right)$$

Use the distributive property to multiply 6x^{2}+12x by \frac{1}{3}.

$$2x^{2}+4x+6x+12=6x-\left(x+2\right)$$

Combine 4x and 6x to get 10x.

$$2x^{2}+10x+12=6x-\left(x+2\right)$$

To find the opposite of x+2, find the opposite of each term.

$$2x^{2}+10x+12=6x-x-2$$

Combine 6x and -x to get 5x.

$$2x^{2}+10x+12=5x-2$$

Subtract 5x from both sides.

$$2x^{2}+10x+12-5x=-2$$

Combine 10x and -5x to get 5x.

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

Add 2 to both sides.

$$2x^{2}+5x+12+2=0$$

Add 12 and 2 to get 14.

$$2x^{2}+5x+14=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 5 for b, and 14 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\times 2\times 14}}{2\times 2}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\times 2\times 14}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-5±\sqrt{25-8\times 14}}{2\times 2}$$

Multiply -8 times 14.

$$x=\frac{-5±\sqrt{25-112}}{2\times 2}$$

Add 25 to -112.

$$x=\frac{-5±\sqrt{-87}}{2\times 2}$$

Take the square root of -87.

$$x=\frac{-5±\sqrt{87}i}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-5±\sqrt{87}i}{4}$$

Now solve the equation x=\frac{-5±\sqrt{87}i}{4} when ± is plus. Add -5 to i\sqrt{87}.

$$x=\frac{-5+\sqrt{87}i}{4}$$

Now solve the equation x=\frac{-5±\sqrt{87}i}{4} when ± is minus. Subtract i\sqrt{87} from -5.

$$x=\frac{-\sqrt{87}i-5}{4}$$

The equation is now solved.

$$x=\frac{-5+\sqrt{87}i}{4} x=\frac{-\sqrt{87}i-5}{4}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -2,0 since division by zero is not defined. Multiply both sides of the equation by 6x\left(x+2\right), the least common multiple of 3,x,2+x,6x.

$$6x\left(x+2\right)\times \frac{1}{3}+6x+12=6x-\left(x+2\right)$$

Use the distributive property to multiply 6x by x+2.

$$\left(6x^{2}+12x\right)\times \frac{1}{3}+6x+12=6x-\left(x+2\right)$$

Use the distributive property to multiply 6x^{2}+12x by \frac{1}{3}.

$$2x^{2}+4x+6x+12=6x-\left(x+2\right)$$

Combine 4x and 6x to get 10x.

$$2x^{2}+10x+12=6x-\left(x+2\right)$$

To find the opposite of x+2, find the opposite of each term.

$$2x^{2}+10x+12=6x-x-2$$

Combine 6x and -x to get 5x.

$$2x^{2}+10x+12=5x-2$$

Subtract 5x from both sides.

$$2x^{2}+10x+12-5x=-2$$

Combine 10x and -5x to get 5x.

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

Subtract 12 from both sides.

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

Subtract 12 from -2 to get -14.

$$2x^{2}+5x=-14$$

Divide both sides by 2.

$$\frac{2x^{2}+5x}{2}=-\frac{14}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{5}{2}x=-\frac{14}{2}$$

Divide -14 by 2.

$$x^{2}+\frac{5}{2}x=-7$$

Divide \frac{5}{2}, the coefficient of the x term, by 2 to get \frac{5}{4}. Then add the square of \frac{5}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{5}{2}x+\left(\frac{5}{4}\right)^{2}=-7+\left(\frac{5}{4}\right)^{2}$$

Square \frac{5}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{5}{2}x+\frac{25}{16}=-7+\frac{25}{16}$$

Add -7 to \frac{25}{16}.

$$x^{2}+\frac{5}{2}x+\frac{25}{16}=-\frac{87}{16}$$

Factor x^{2}+\frac{5}{2}x+\frac{25}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{5}{4}\right)^{2}=-\frac{87}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{5}{4}\right)^{2}}=\sqrt{-\frac{87}{16}}$$

Simplify.

$$x+\frac{5}{4}=\frac{\sqrt{87}i}{4} x+\frac{5}{4}=-\frac{\sqrt{87}i}{4}$$

Subtract \frac{5}{4} from both sides of the equation.

$$x=\frac{-5+\sqrt{87}i}{4} x=\frac{-\sqrt{87}i-5}{4}$$