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

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

Solve for x

$$x=-\frac{1}{6}\approx -0.166666667$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -1,0,1 since division by zero is not defined. Multiply both sides of the equation by 6x\left(x-1\right)\left(x+1\right), the least common multiple of x^{2}-1,2x^{2}-2x,3x^{2}+3x.

$$6xx-\left(3x+3\right)=2x-2$$

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

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

To find the opposite of 3x+3, find the opposite of each term.

$$6x^{2}-3x-3=2x-2$$

Subtract 2x from both sides.

$$6x^{2}-3x-3-2x=-2$$

Combine -3x and -2x to get -5x.

$$6x^{2}-5x-3=-2$$

Add 2 to both sides.

$$6x^{2}-5x-3+2=0$$

Add -3 and 2 to get -1.

$$6x^{2}-5x-1=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 6x^{2}+ax+bx-1. To find a and b, set up a system to be solved.

$$a+b=-5 ab=6\left(-1\right)=-6$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -6.

$$1,-6 2,-3$$

Calculate the sum for each pair.

$$1-6=-5 2-3=-1$$

The solution is the pair that gives sum -5.

$$a=-6 b=1$$

Rewrite 6x^{2}-5x-1 as \left(6x^{2}-6x\right)+\left(x-1\right).

$$\left(6x^{2}-6x\right)+\left(x-1\right)$$

Factor out 6x in 6x^{2}-6x.

$$6x\left(x-1\right)+x-1$$

Factor out common term x-1 by using distributive property.

$$\left(x-1\right)\left(6x+1\right)$$

To find equation solutions, solve x-1=0 and 6x+1=0.

$$x=1 x=-\frac{1}{6}$$

Variable x cannot be equal to 1.

$$x=-\frac{1}{6}$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -1,0,1 since division by zero is not defined. Multiply both sides of the equation by 6x\left(x-1\right)\left(x+1\right), the least common multiple of x^{2}-1,2x^{2}-2x,3x^{2}+3x.

$$6xx-\left(3x+3\right)=2x-2$$

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

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

To find the opposite of 3x+3, find the opposite of each term.

$$6x^{2}-3x-3=2x-2$$

Subtract 2x from both sides.

$$6x^{2}-3x-3-2x=-2$$

Combine -3x and -2x to get -5x.

$$6x^{2}-5x-3=-2$$

Add 2 to both sides.

$$6x^{2}-5x-3+2=0$$

Add -3 and 2 to get -1.

$$6x^{2}-5x-1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, -5 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 6\left(-1\right)}}{2\times 6}$$

Square -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\times 6\left(-1\right)}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-\left(-5\right)±\sqrt{25-24\left(-1\right)}}{2\times 6}$$

Multiply -24 times -1.

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

Add 25 to 24.

$$x=\frac{-\left(-5\right)±\sqrt{49}}{2\times 6}$$

Take the square root of 49.

$$x=\frac{-\left(-5\right)±7}{2\times 6}$$

The opposite of -5 is 5.

$$x=\frac{5±7}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{5±7}{12}$$

Now solve the equation x=\frac{5±7}{12} when ± is plus. Add 5 to 7.

$$x=\frac{12}{12}$$

Divide 12 by 12.

$$x=1$$

Now solve the equation x=\frac{5±7}{12} when ± is minus. Subtract 7 from 5.

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

Reduce the fraction \frac{-2}{12} to lowest terms by extracting and canceling out 2.

$$x=-\frac{1}{6}$$

The equation is now solved.

$$x=1 x=-\frac{1}{6}$$

Variable x cannot be equal to 1.

$$x=-\frac{1}{6}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -1,0,1 since division by zero is not defined. Multiply both sides of the equation by 6x\left(x-1\right)\left(x+1\right), the least common multiple of x^{2}-1,2x^{2}-2x,3x^{2}+3x.

$$6xx-\left(3x+3\right)=2x-2$$

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

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

To find the opposite of 3x+3, find the opposite of each term.

$$6x^{2}-3x-3=2x-2$$

Subtract 2x from both sides.

$$6x^{2}-3x-3-2x=-2$$

Combine -3x and -2x to get -5x.

$$6x^{2}-5x-3=-2$$

Add 3 to both sides.

$$6x^{2}-5x=-2+3$$

Add -2 and 3 to get 1.

$$6x^{2}-5x=1$$

Divide both sides by 6.

$$\frac{6x^{2}-5x}{6}=\frac{1}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}-\frac{5}{6}x=\frac{1}{6}$$

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.

$$x^{2}-\frac{5}{6}x+\left(-\frac{5}{12}\right)^{2}=\frac{1}{6}+\left(-\frac{5}{12}\right)^{2}$$

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

$$x^{2}-\frac{5}{6}x+\frac{25}{144}=\frac{1}{6}+\frac{25}{144}$$

Add \frac{1}{6} to \frac{25}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{5}{6}x+\frac{25}{144}=\frac{49}{144}$$

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}.

$$\left(x-\frac{5}{12}\right)^{2}=\frac{49}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{12}\right)^{2}}=\sqrt{\frac{49}{144}}$$

Simplify.

$$x-\frac{5}{12}=\frac{7}{12} x-\frac{5}{12}=-\frac{7}{12}$$

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

$$x=1 x=-\frac{1}{6}$$

Variable x cannot be equal to 1.

$$x=-\frac{1}{6}$$