Solve 3x(x+2)/x^2-1+1/2x=9+x^3/2/x^2-1-frac{1}{2(x^2-1)}

$$\frac { 3 x ( x + 2 ) } { x ^ { 2 } - 1 } + \frac { 1 } { 2 } x = \frac { 9 + \frac { x ^ { 3 } } { 2 } } { x ^ { 2 } - 1 } - \frac { 1 } { 2 ( x ^ { 2 } - 1 ) }$$

Solve for x

$$x = -\frac{17}{6} = -2\frac{5}{6} \approx -2.833333333$$

Steps Using Factoring By Grouping

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

$$2\times 3x\left(x+2\right)+\frac{1}{2}x\times 2\left(x-1\right)\left(x+1\right)=2\left(9+\frac{x^{3}}{2}\right)-1$$

Multiply 2 and 3 to get 6.

$$6x\left(x+2\right)+\frac{1}{2}x\times 2\left(x-1\right)\left(x+1\right)=2\left(9+\frac{x^{3}}{2}\right)-1$$

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

$$6x^{2}+12x+\frac{1}{2}x\times 2\left(x-1\right)\left(x+1\right)=2\left(9+\frac{x^{3}}{2}\right)-1$$

Multiply \frac{1}{2} and 2 to get 1.

$$6x^{2}+12x+x\left(x-1\right)\left(x+1\right)=2\left(9+\frac{x^{3}}{2}\right)-1$$

Use the distributive property to multiply x by x-1.

$$6x^{2}+12x+\left(x^{2}-x\right)\left(x+1\right)=2\left(9+\frac{x^{3}}{2}\right)-1$$

Use the distributive property to multiply x^{2}-x by x+1 and combine like terms.

$$6x^{2}+12x+x^{3}-x=2\left(9+\frac{x^{3}}{2}\right)-1$$

Combine 12x and -x to get 11x.

$$6x^{2}+11x+x^{3}=2\left(9+\frac{x^{3}}{2}\right)-1$$

Use the distributive property to multiply 2 by 9+\frac{x^{3}}{2}.

$$6x^{2}+11x+x^{3}=18+2\times \frac{x^{3}}{2}-1$$

Express 2\times \frac{x^{3}}{2} as a single fraction.

$$6x^{2}+11x+x^{3}=18+\frac{2x^{3}}{2}-1$$

Cancel out 2 and 2.

$$6x^{2}+11x+x^{3}=18+x^{3}-1$$

Subtract 1 from 18 to get 17.

$$6x^{2}+11x+x^{3}=17+x^{3}$$

Subtract 17 from both sides.

$$6x^{2}+11x+x^{3}-17=x^{3}$$

Subtract x^{3} from both sides.

$$6x^{2}+11x+x^{3}-17-x^{3}=0$$

Combine x^{3} and -x^{3} to get 0.

$$6x^{2}+11x-17=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 6x^{2}+ax+bx-17. To find a and b, set up a system to be solved.

$$a+b=11 ab=6\left(-17\right)=-102$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -102.

$$-1,102 -2,51 -3,34 -6,17$$

Calculate the sum for each pair.

$$-1+102=101 -2+51=49 -3+34=31 -6+17=11$$

The solution is the pair that gives sum 11.

$$a=-6 b=17$$

Rewrite 6x^{2}+11x-17 as \left(6x^{2}-6x\right)+\left(17x-17\right).

$$\left(6x^{2}-6x\right)+\left(17x-17\right)$$

Factor out 6x in the first and 17 in the second group.

$$6x\left(x-1\right)+17\left(x-1\right)$$

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

$$\left(x-1\right)\left(6x+17\right)$$

To find equation solutions, solve x-1=0 and 6x+17=0.

$$x=1 x=-\frac{17}{6}$$

Variable x cannot be equal to 1.

$$x=-\frac{17}{6}$$

Steps Using the Quadratic Formula

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

$$2\times 3x\left(x+2\right)+\frac{1}{2}x\times 2\left(x-1\right)\left(x+1\right)=2\left(9+\frac{x^{3}}{2}\right)-1$$

Multiply 2 and 3 to get 6.

$$6x\left(x+2\right)+\frac{1}{2}x\times 2\left(x-1\right)\left(x+1\right)=2\left(9+\frac{x^{3}}{2}\right)-1$$

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

$$6x^{2}+12x+\frac{1}{2}x\times 2\left(x-1\right)\left(x+1\right)=2\left(9+\frac{x^{3}}{2}\right)-1$$

Multiply \frac{1}{2} and 2 to get 1.

$$6x^{2}+12x+x\left(x-1\right)\left(x+1\right)=2\left(9+\frac{x^{3}}{2}\right)-1$$

Use the distributive property to multiply x by x-1.

$$6x^{2}+12x+\left(x^{2}-x\right)\left(x+1\right)=2\left(9+\frac{x^{3}}{2}\right)-1$$

Use the distributive property to multiply x^{2}-x by x+1 and combine like terms.

$$6x^{2}+12x+x^{3}-x=2\left(9+\frac{x^{3}}{2}\right)-1$$

Combine 12x and -x to get 11x.

$$6x^{2}+11x+x^{3}=2\left(9+\frac{x^{3}}{2}\right)-1$$

Use the distributive property to multiply 2 by 9+\frac{x^{3}}{2}.

$$6x^{2}+11x+x^{3}=18+2\times \frac{x^{3}}{2}-1$$

Express 2\times \frac{x^{3}}{2} as a single fraction.

$$6x^{2}+11x+x^{3}=18+\frac{2x^{3}}{2}-1$$

Cancel out 2 and 2.

$$6x^{2}+11x+x^{3}=18+x^{3}-1$$

Subtract 1 from 18 to get 17.

$$6x^{2}+11x+x^{3}=17+x^{3}$$

Subtract 17 from both sides.

$$6x^{2}+11x+x^{3}-17=x^{3}$$

Subtract x^{3} from both sides.

$$6x^{2}+11x+x^{3}-17-x^{3}=0$$

Combine x^{3} and -x^{3} to get 0.

$$6x^{2}+11x-17=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, 11 for b, and -17 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-11±\sqrt{11^{2}-4\times 6\left(-17\right)}}{2\times 6}$$

Square 11.

$$x=\frac{-11±\sqrt{121-4\times 6\left(-17\right)}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-11±\sqrt{121-24\left(-17\right)}}{2\times 6}$$

Multiply -24 times -17.

$$x=\frac{-11±\sqrt{121+408}}{2\times 6}$$

Add 121 to 408.

$$x=\frac{-11±\sqrt{529}}{2\times 6}$$

Take the square root of 529.

$$x=\frac{-11±23}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{-11±23}{12}$$

Now solve the equation x=\frac{-11±23}{12} when ± is plus. Add -11 to 23.

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

Divide 12 by 12.

$$x=1$$

Now solve the equation x=\frac{-11±23}{12} when ± is minus. Subtract 23 from -11.

$$x=-\frac{34}{12}$$

Reduce the fraction \frac{-34}{12} to lowest terms by extracting and canceling out 2.

$$x=-\frac{17}{6}$$

The equation is now solved.

$$x=1 x=-\frac{17}{6}$$

Variable x cannot be equal to 1.

$$x=-\frac{17}{6}$$

Steps for Completing the Square

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

$$2\times 3x\left(x+2\right)+\frac{1}{2}x\times 2\left(x-1\right)\left(x+1\right)=2\left(9+\frac{x^{3}}{2}\right)-1$$

Multiply 2 and 3 to get 6.

$$6x\left(x+2\right)+\frac{1}{2}x\times 2\left(x-1\right)\left(x+1\right)=2\left(9+\frac{x^{3}}{2}\right)-1$$

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

$$6x^{2}+12x+\frac{1}{2}x\times 2\left(x-1\right)\left(x+1\right)=2\left(9+\frac{x^{3}}{2}\right)-1$$

Multiply \frac{1}{2} and 2 to get 1.

$$6x^{2}+12x+x\left(x-1\right)\left(x+1\right)=2\left(9+\frac{x^{3}}{2}\right)-1$$

Use the distributive property to multiply x by x-1.

$$6x^{2}+12x+\left(x^{2}-x\right)\left(x+1\right)=2\left(9+\frac{x^{3}}{2}\right)-1$$

Use the distributive property to multiply x^{2}-x by x+1 and combine like terms.

$$6x^{2}+12x+x^{3}-x=2\left(9+\frac{x^{3}}{2}\right)-1$$

Combine 12x and -x to get 11x.

$$6x^{2}+11x+x^{3}=2\left(9+\frac{x^{3}}{2}\right)-1$$

Use the distributive property to multiply 2 by 9+\frac{x^{3}}{2}.

$$6x^{2}+11x+x^{3}=18+2\times \frac{x^{3}}{2}-1$$

Express 2\times \frac{x^{3}}{2} as a single fraction.

$$6x^{2}+11x+x^{3}=18+\frac{2x^{3}}{2}-1$$

Cancel out 2 and 2.

$$6x^{2}+11x+x^{3}=18+x^{3}-1$$

Subtract 1 from 18 to get 17.

$$6x^{2}+11x+x^{3}=17+x^{3}$$

Subtract x^{3} from both sides.

$$6x^{2}+11x+x^{3}-x^{3}=17$$

Combine x^{3} and -x^{3} to get 0.

$$6x^{2}+11x=17$$

Divide both sides by 6.

$$\frac{6x^{2}+11x}{6}=\frac{17}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}+\frac{11}{6}x=\frac{17}{6}$$

Divide \frac{11}{6}, the coefficient of the x term, by 2 to get \frac{11}{12}. Then add the square of \frac{11}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{11}{6}x+\left(\frac{11}{12}\right)^{2}=\frac{17}{6}+\left(\frac{11}{12}\right)^{2}$$

Square \frac{11}{12} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{11}{6}x+\frac{121}{144}=\frac{17}{6}+\frac{121}{144}$$

Add \frac{17}{6} to \frac{121}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{11}{6}x+\frac{121}{144}=\frac{529}{144}$$

Factor x^{2}+\frac{11}{6}x+\frac{121}{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}.

$$\left(x+\frac{11}{12}\right)^{2}=\frac{529}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{11}{12}\right)^{2}}=\sqrt{\frac{529}{144}}$$

Simplify.

$$x+\frac{11}{12}=\frac{23}{12} x+\frac{11}{12}=-\frac{23}{12}$$

Subtract \frac{11}{12} from both sides of the equation.

$$x=1 x=-\frac{17}{6}$$

Variable x cannot be equal to 1.

$$x=-\frac{17}{6}$$