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

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

Solve for x

$$x=5$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -2,-1,1,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x-1\right)\left(x+1\right)\left(x+2\right), the least common multiple of x^{2}-3x+2,x^{2}+3x+2,x^{2}-4.

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

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

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

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

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

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

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

Combine 2x^{2} and x^{2} to get 3x^{2}.

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

Combine 6x and -3x to get 3x.

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

Add 4 and 2 to get 6.

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

Use the distributive property to multiply x^{2}-1 by 4.

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

Subtract 4x^{2} from both sides.

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

Combine 3x^{2} and -4x^{2} to get -x^{2}.

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

Add 4 to both sides.

$$-x^{2}+3x+6+4=0$$

Add 6 and 4 to get 10.

$$-x^{2}+3x+10=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+10. To find a and b, set up a system to be solved.

$$a+b=3 ab=-10=-10$$

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 -10.

$$-1,10 -2,5$$

Calculate the sum for each pair.

$$-1+10=9 -2+5=3$$

The solution is the pair that gives sum 3.

$$a=5 b=-2$$

Rewrite -x^{2}+3x+10 as \left(-x^{2}+5x\right)+\left(-2x+10\right).

$$\left(-x^{2}+5x\right)+\left(-2x+10\right)$$

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

$$-x\left(x-5\right)-2\left(x-5\right)$$

Factor out common term x-5 by using distributive property.

$$\left(x-5\right)\left(-x-2\right)$$

To find equation solutions, solve x-5=0 and -x-2=0.

$$x=5 x=-2$$

Variable x cannot be equal to -2.

$$x=5$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -2,-1,1,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x-1\right)\left(x+1\right)\left(x+2\right), the least common multiple of x^{2}-3x+2,x^{2}+3x+2,x^{2}-4.

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

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

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

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

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

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

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

Combine 2x^{2} and x^{2} to get 3x^{2}.

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

Combine 6x and -3x to get 3x.

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

Add 4 and 2 to get 6.

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

Use the distributive property to multiply x^{2}-1 by 4.

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

Subtract 4x^{2} from both sides.

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

Combine 3x^{2} and -4x^{2} to get -x^{2}.

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

Add 4 to both sides.

$$-x^{2}+3x+6+4=0$$

Add 6 and 4 to get 10.

$$-x^{2}+3x+10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 3 for b, and 10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3±\sqrt{3^{2}-4\left(-1\right)\times 10}}{2\left(-1\right)}$$

Square 3.

$$x=\frac{-3±\sqrt{9-4\left(-1\right)\times 10}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-3±\sqrt{9+4\times 10}}{2\left(-1\right)}$$

Multiply 4 times 10.

$$x=\frac{-3±\sqrt{9+40}}{2\left(-1\right)}$$

Add 9 to 40.

$$x=\frac{-3±\sqrt{49}}{2\left(-1\right)}$$

Take the square root of 49.

$$x=\frac{-3±7}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-3±7}{-2}$$

Now solve the equation x=\frac{-3±7}{-2} when ± is plus. Add -3 to 7.

$$x=\frac{4}{-2}$$

Divide 4 by -2.

$$x=-2$$

Now solve the equation x=\frac{-3±7}{-2} when ± is minus. Subtract 7 from -3.

$$x=-\frac{10}{-2}$$

Divide -10 by -2.

$$x=5$$

The equation is now solved.

$$x=-2 x=5$$

Variable x cannot be equal to -2.

$$x=5$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -2,-1,1,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x-1\right)\left(x+1\right)\left(x+2\right), the least common multiple of x^{2}-3x+2,x^{2}+3x+2,x^{2}-4.

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

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

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

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

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

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

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

Combine 2x^{2} and x^{2} to get 3x^{2}.

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

Combine 6x and -3x to get 3x.

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

Add 4 and 2 to get 6.

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

Use the distributive property to multiply x^{2}-1 by 4.

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

Subtract 4x^{2} from both sides.

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

Combine 3x^{2} and -4x^{2} to get -x^{2}.

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

Subtract 6 from both sides.

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

Subtract 6 from -4 to get -10.

$$-x^{2}+3x=-10$$

Divide both sides by -1.

$$\frac{-x^{2}+3x}{-1}=-\frac{10}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{3}{-1}x=-\frac{10}{-1}$$

Divide 3 by -1.

$$x^{2}-3x=-\frac{10}{-1}$$

Divide -10 by -1.

$$x^{2}-3x=10$$

Divide -3, the coefficient of the x term, by 2 to get -\frac{3}{2}. Then add the square of -\frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-3x+\left(-\frac{3}{2}\right)^{2}=10+\left(-\frac{3}{2}\right)^{2}$$

Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-3x+\frac{9}{4}=10+\frac{9}{4}$$

Add 10 to \frac{9}{4}.

$$x^{2}-3x+\frac{9}{4}=\frac{49}{4}$$

Factor x^{2}-3x+\frac{9}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{49}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{49}{4}}$$

Simplify.

$$x-\frac{3}{2}=\frac{7}{2} x-\frac{3}{2}=-\frac{7}{2}$$

Add \frac{3}{2} to both sides of the equation.

$$x=5 x=-2$$

Variable x cannot be equal to -2.

$$x=5$$