Solve 2/x^2-1-4/x^2-3x+2=1/3x^2+6x+3

$$\frac { 2 } { x ^ { 2 } - 1 } - \frac { 4 } { x ^ { 2 } - 3 x + 2 } = \frac { 1 } { 3 x ^ { 2 } + 6 x + 3 }$$

Solve for x

$$x = -\frac{13}{7} = -1\frac{6}{7} \approx -1.857142857$$

$$x=-2$$

Steps Using Factoring By Grouping

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

$$3\left(x-2\right)\left(x+1\right)\times 2-3\left(x+1\right)^{2}\times 4=\left(x-2\right)\left(x-1\right)$$

Use the distributive property to multiply 3 by x-2.

$$\left(3x-6\right)\left(x+1\right)\times 2-3\left(x+1\right)^{2}\times 4=\left(x-2\right)\left(x-1\right)$$

Use the distributive property to multiply 3x-6 by x+1 and combine like terms.

$$\left(3x^{2}-3x-6\right)\times 2-3\left(x+1\right)^{2}\times 4=\left(x-2\right)\left(x-1\right)$$

Use the distributive property to multiply 3x^{2}-3x-6 by 2.

$$6x^{2}-6x-12-3\left(x+1\right)^{2}\times 4=\left(x-2\right)\left(x-1\right)$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

$$6x^{2}-6x-12-3\left(x^{2}+2x+1\right)\times 4=\left(x-2\right)\left(x-1\right)$$

Multiply 3 and 4 to get 12.

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

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

$$6x^{2}-6x-12-\left(12x^{2}+24x+12\right)=\left(x-2\right)\left(x-1\right)$$

To find the opposite of 12x^{2}+24x+12, find the opposite of each term.

$$6x^{2}-6x-12-12x^{2}-24x-12=\left(x-2\right)\left(x-1\right)$$

Combine 6x^{2} and -12x^{2} to get -6x^{2}.

$$-6x^{2}-6x-12-24x-12=\left(x-2\right)\left(x-1\right)$$

Combine -6x and -24x to get -30x.

$$-6x^{2}-30x-12-12=\left(x-2\right)\left(x-1\right)$$

Subtract 12 from -12 to get -24.

$$-6x^{2}-30x-24=\left(x-2\right)\left(x-1\right)$$

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

$$-6x^{2}-30x-24=x^{2}-3x+2$$

Subtract x^{2} from both sides.

$$-6x^{2}-30x-24-x^{2}=-3x+2$$

Combine -6x^{2} and -x^{2} to get -7x^{2}.

$$-7x^{2}-30x-24=-3x+2$$

Add 3x to both sides.

$$-7x^{2}-30x-24+3x=2$$

Combine -30x and 3x to get -27x.

$$-7x^{2}-27x-24=2$$

Subtract 2 from both sides.

$$-7x^{2}-27x-24-2=0$$

Subtract 2 from -24 to get -26.

$$-7x^{2}-27x-26=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -7x^{2}+ax+bx-26. To find a and b, set up a system to be solved.

$$a+b=-27 ab=-7\left(-26\right)=182$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 182.

$$-1,-182 -2,-91 -7,-26 -13,-14$$

Calculate the sum for each pair.

$$-1-182=-183 -2-91=-93 -7-26=-33 -13-14=-27$$

The solution is the pair that gives sum -27.

$$a=-13 b=-14$$

Rewrite -7x^{2}-27x-26 as \left(-7x^{2}-13x\right)+\left(-14x-26\right).

$$\left(-7x^{2}-13x\right)+\left(-14x-26\right)$$

Factor out -x in the first and -2 in the second group.

$$-x\left(7x+13\right)-2\left(7x+13\right)$$

Factor out common term 7x+13 by using distributive property.

$$\left(7x+13\right)\left(-x-2\right)$$

To find equation solutions, solve 7x+13=0 and -x-2=0.

$$x=-\frac{13}{7} x=-2$$

Steps Using the Quadratic Formula

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

$$3\left(x-2\right)\left(x+1\right)\times 2-3\left(x+1\right)^{2}\times 4=\left(x-2\right)\left(x-1\right)$$

Use the distributive property to multiply 3 by x-2.

$$\left(3x-6\right)\left(x+1\right)\times 2-3\left(x+1\right)^{2}\times 4=\left(x-2\right)\left(x-1\right)$$

Use the distributive property to multiply 3x-6 by x+1 and combine like terms.

$$\left(3x^{2}-3x-6\right)\times 2-3\left(x+1\right)^{2}\times 4=\left(x-2\right)\left(x-1\right)$$

Use the distributive property to multiply 3x^{2}-3x-6 by 2.

$$6x^{2}-6x-12-3\left(x+1\right)^{2}\times 4=\left(x-2\right)\left(x-1\right)$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

$$6x^{2}-6x-12-3\left(x^{2}+2x+1\right)\times 4=\left(x-2\right)\left(x-1\right)$$

Multiply 3 and 4 to get 12.

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

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

$$6x^{2}-6x-12-\left(12x^{2}+24x+12\right)=\left(x-2\right)\left(x-1\right)$$

To find the opposite of 12x^{2}+24x+12, find the opposite of each term.

$$6x^{2}-6x-12-12x^{2}-24x-12=\left(x-2\right)\left(x-1\right)$$

Combine 6x^{2} and -12x^{2} to get -6x^{2}.

$$-6x^{2}-6x-12-24x-12=\left(x-2\right)\left(x-1\right)$$

Combine -6x and -24x to get -30x.

$$-6x^{2}-30x-12-12=\left(x-2\right)\left(x-1\right)$$

Subtract 12 from -12 to get -24.

$$-6x^{2}-30x-24=\left(x-2\right)\left(x-1\right)$$

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

$$-6x^{2}-30x-24=x^{2}-3x+2$$

Subtract x^{2} from both sides.

$$-6x^{2}-30x-24-x^{2}=-3x+2$$

Combine -6x^{2} and -x^{2} to get -7x^{2}.

$$-7x^{2}-30x-24=-3x+2$$

Add 3x to both sides.

$$-7x^{2}-30x-24+3x=2$$

Combine -30x and 3x to get -27x.

$$-7x^{2}-27x-24=2$$

Subtract 2 from both sides.

$$-7x^{2}-27x-24-2=0$$

Subtract 2 from -24 to get -26.

$$-7x^{2}-27x-26=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -7 for a, -27 for b, and -26 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-27\right)±\sqrt{\left(-27\right)^{2}-4\left(-7\right)\left(-26\right)}}{2\left(-7\right)}$$

Square -27.

$$x=\frac{-\left(-27\right)±\sqrt{729-4\left(-7\right)\left(-26\right)}}{2\left(-7\right)}$$

Multiply -4 times -7.

$$x=\frac{-\left(-27\right)±\sqrt{729+28\left(-26\right)}}{2\left(-7\right)}$$

Multiply 28 times -26.

$$x=\frac{-\left(-27\right)±\sqrt{729-728}}{2\left(-7\right)}$$

Add 729 to -728.

$$x=\frac{-\left(-27\right)±\sqrt{1}}{2\left(-7\right)}$$

Take the square root of 1.

$$x=\frac{-\left(-27\right)±1}{2\left(-7\right)}$$

The opposite of -27 is 27.

$$x=\frac{27±1}{2\left(-7\right)}$$

Multiply 2 times -7.

$$x=\frac{27±1}{-14}$$

Now solve the equation x=\frac{27±1}{-14} when ± is plus. Add 27 to 1.

$$x=\frac{28}{-14}$$

Divide 28 by -14.

$$x=-2$$

Now solve the equation x=\frac{27±1}{-14} when ± is minus. Subtract 1 from 27.

$$x=\frac{26}{-14}$$

Reduce the fraction \frac{26}{-14} to lowest terms by extracting and canceling out 2.

$$x=-\frac{13}{7}$$

The equation is now solved.

$$x=-2 x=-\frac{13}{7}$$

Steps for Completing the Square

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

$$3\left(x-2\right)\left(x+1\right)\times 2-3\left(x+1\right)^{2}\times 4=\left(x-2\right)\left(x-1\right)$$

Use the distributive property to multiply 3 by x-2.

$$\left(3x-6\right)\left(x+1\right)\times 2-3\left(x+1\right)^{2}\times 4=\left(x-2\right)\left(x-1\right)$$

Use the distributive property to multiply 3x-6 by x+1 and combine like terms.

$$\left(3x^{2}-3x-6\right)\times 2-3\left(x+1\right)^{2}\times 4=\left(x-2\right)\left(x-1\right)$$

Use the distributive property to multiply 3x^{2}-3x-6 by 2.

$$6x^{2}-6x-12-3\left(x+1\right)^{2}\times 4=\left(x-2\right)\left(x-1\right)$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

$$6x^{2}-6x-12-3\left(x^{2}+2x+1\right)\times 4=\left(x-2\right)\left(x-1\right)$$

Multiply 3 and 4 to get 12.

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

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

$$6x^{2}-6x-12-\left(12x^{2}+24x+12\right)=\left(x-2\right)\left(x-1\right)$$

To find the opposite of 12x^{2}+24x+12, find the opposite of each term.

$$6x^{2}-6x-12-12x^{2}-24x-12=\left(x-2\right)\left(x-1\right)$$

Combine 6x^{2} and -12x^{2} to get -6x^{2}.

$$-6x^{2}-6x-12-24x-12=\left(x-2\right)\left(x-1\right)$$

Combine -6x and -24x to get -30x.

$$-6x^{2}-30x-12-12=\left(x-2\right)\left(x-1\right)$$

Subtract 12 from -12 to get -24.

$$-6x^{2}-30x-24=\left(x-2\right)\left(x-1\right)$$

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

$$-6x^{2}-30x-24=x^{2}-3x+2$$

Subtract x^{2} from both sides.

$$-6x^{2}-30x-24-x^{2}=-3x+2$$

Combine -6x^{2} and -x^{2} to get -7x^{2}.

$$-7x^{2}-30x-24=-3x+2$$

Add 3x to both sides.

$$-7x^{2}-30x-24+3x=2$$

Combine -30x and 3x to get -27x.

$$-7x^{2}-27x-24=2$$

Add 24 to both sides.

$$-7x^{2}-27x=2+24$$

Add 2 and 24 to get 26.

$$-7x^{2}-27x=26$$

Divide both sides by -7.

$$\frac{-7x^{2}-27x}{-7}=\frac{26}{-7}$$

Dividing by -7 undoes the multiplication by -7.

$$x^{2}+\left(-\frac{27}{-7}\right)x=\frac{26}{-7}$$

Divide -27 by -7.

$$x^{2}+\frac{27}{7}x=\frac{26}{-7}$$

Divide 26 by -7.

$$x^{2}+\frac{27}{7}x=-\frac{26}{7}$$

Divide \frac{27}{7}, the coefficient of the x term, by 2 to get \frac{27}{14}. Then add the square of \frac{27}{14} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{27}{7}x+\left(\frac{27}{14}\right)^{2}=-\frac{26}{7}+\left(\frac{27}{14}\right)^{2}$$

Square \frac{27}{14} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{27}{7}x+\frac{729}{196}=-\frac{26}{7}+\frac{729}{196}$$

Add -\frac{26}{7} to \frac{729}{196} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{27}{7}x+\frac{729}{196}=\frac{1}{196}$$

Factor x^{2}+\frac{27}{7}x+\frac{729}{196}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{27}{14}\right)^{2}=\frac{1}{196}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{27}{14}\right)^{2}}=\sqrt{\frac{1}{196}}$$

Simplify.

$$x+\frac{27}{14}=\frac{1}{14} x+\frac{27}{14}=-\frac{1}{14}$$

Subtract \frac{27}{14} from both sides of the equation.

$$x=-\frac{13}{7} x=-2$$