Solve x/x+2-3=8/x^2-4

$$\frac { x } { x + 2 } - 3 = \frac { 8 } { x ^ { 2 } - 4 }$$

Solve for x

$$x=1$$

Steps Using Factoring By Grouping

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

$$\left(x-2\right)x+\left(x-2\right)\left(x+2\right)\left(-3\right)=8$$

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

$$x^{2}-2x+\left(x-2\right)\left(x+2\right)\left(-3\right)=8$$

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

$$x^{2}-2x+\left(x^{2}-4\right)\left(-3\right)=8$$

Use the distributive property to multiply x^{2}-4 by -3.

$$x^{2}-2x-3x^{2}+12=8$$

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

$$-2x^{2}-2x+12=8$$

Subtract 8 from both sides.

$$-2x^{2}-2x+12-8=0$$

Subtract 8 from 12 to get 4.

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

Divide both sides by 2.

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

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

$$a+b=-1 ab=-2=-2$$

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. The only such pair is the system solution.

$$a=1 b=-2$$

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

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

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

$$x\left(-x+1\right)+2\left(-x+1\right)$$

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

$$\left(-x+1\right)\left(x+2\right)$$

To find equation solutions, solve -x+1=0 and x+2=0.

$$x=1 x=-2$$

Variable x cannot be equal to -2.

$$x=1$$

Steps Using the Quadratic Formula

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

$$\left(x-2\right)x+\left(x-2\right)\left(x+2\right)\left(-3\right)=8$$

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

$$x^{2}-2x+\left(x-2\right)\left(x+2\right)\left(-3\right)=8$$

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

$$x^{2}-2x+\left(x^{2}-4\right)\left(-3\right)=8$$

Use the distributive property to multiply x^{2}-4 by -3.

$$x^{2}-2x-3x^{2}+12=8$$

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

$$-2x^{2}-2x+12=8$$

Subtract 8 from both sides.

$$-2x^{2}-2x+12-8=0$$

Subtract 8 from 12 to get 4.

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

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

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

Square -2.

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

Multiply -4 times -2.

$$x=\frac{-\left(-2\right)±\sqrt{4+8\times 4}}{2\left(-2\right)}$$

Multiply 8 times 4.

$$x=\frac{-\left(-2\right)±\sqrt{4+32}}{2\left(-2\right)}$$

Add 4 to 32.

$$x=\frac{-\left(-2\right)±\sqrt{36}}{2\left(-2\right)}$$

Take the square root of 36.

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

The opposite of -2 is 2.

$$x=\frac{2±6}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{2±6}{-4}$$

Now solve the equation x=\frac{2±6}{-4} when ± is plus. Add 2 to 6.

$$x=\frac{8}{-4}$$

Divide 8 by -4.

$$x=-2$$

Now solve the equation x=\frac{2±6}{-4} when ± is minus. Subtract 6 from 2.

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

Divide -4 by -4.

$$x=1$$

The equation is now solved.

$$x=-2 x=1$$

Variable x cannot be equal to -2.

$$x=1$$

Steps for Completing the Square

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

$$\left(x-2\right)x+\left(x-2\right)\left(x+2\right)\left(-3\right)=8$$

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

$$x^{2}-2x+\left(x-2\right)\left(x+2\right)\left(-3\right)=8$$

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

$$x^{2}-2x+\left(x^{2}-4\right)\left(-3\right)=8$$

Use the distributive property to multiply x^{2}-4 by -3.

$$x^{2}-2x-3x^{2}+12=8$$

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

$$-2x^{2}-2x+12=8$$

Subtract 12 from both sides.

$$-2x^{2}-2x=8-12$$

Subtract 12 from 8 to get -4.

$$-2x^{2}-2x=-4$$

Divide both sides by -2.

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

Dividing by -2 undoes the multiplication by -2.

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

Divide -2 by -2.

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

Divide -4 by -2.

$$x^{2}+x=2$$

Divide 1, the coefficient of the x term, by 2 to get \frac{1}{2}. Then add the square of \frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

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

Add 2 to \frac{1}{4}.

$$x^{2}+x+\frac{1}{4}=\frac{9}{4}$$

Factor x^{2}+x+\frac{1}{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}.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{9}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Simplify.

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

Subtract \frac{1}{2} from both sides of the equation.

$$x=1 x=-2$$

Variable x cannot be equal to -2.

$$x=1$$