Solve 3x/x+2+2x+1/2-x=2x^2+16/x^2-4

$$\frac{ 3x }{ x+2 } + \frac{ 2x+1 }{ 2-x } = \frac{ 2 { x }^{ 2 } +16 }{ { x }^{ 2 } -4 }$$

Solve for x

$$x=-9$$

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+2,2-x,x^{2}-4.

$$\left(x-2\right)\times 3x-\left(2+x\right)\left(2x+1\right)=2x^{2}+16$$

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

$$\left(3x-6\right)x-\left(2+x\right)\left(2x+1\right)=2x^{2}+16$$

Use the distributive property to multiply 3x-6 by x.

$$3x^{2}-6x-\left(2+x\right)\left(2x+1\right)=2x^{2}+16$$

Use the distributive property to multiply -1 by 2+x.

$$3x^{2}-6x+\left(-2-x\right)\left(2x+1\right)=2x^{2}+16$$

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

$$3x^{2}-6x-5x-2-2x^{2}=2x^{2}+16$$

Combine -6x and -5x to get -11x.

$$3x^{2}-11x-2-2x^{2}=2x^{2}+16$$

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

$$x^{2}-11x-2=2x^{2}+16$$

Subtract 2x^{2} from both sides.

$$x^{2}-11x-2-2x^{2}=16$$

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

$$-x^{2}-11x-2=16$$

Subtract 16 from both sides.

$$-x^{2}-11x-2-16=0$$

Subtract 16 from -2 to get -18.

$$-x^{2}-11x-18=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-18. To find a and b, set up a system to be solved.

$$a+b=-11 ab=-\left(-18\right)=18$$

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 18.

$$-1,-18 -2,-9 -3,-6$$

Calculate the sum for each pair.

$$-1-18=-19 -2-9=-11 -3-6=-9$$

The solution is the pair that gives sum -11.

$$a=-2 b=-9$$

Rewrite -x^{2}-11x-18 as \left(-x^{2}-2x\right)+\left(-9x-18\right).

$$\left(-x^{2}-2x\right)+\left(-9x-18\right)$$

Factor out x in the first and 9 in the second group.

$$x\left(-x-2\right)+9\left(-x-2\right)$$

Factor out common term -x-2 by using distributive property.

$$\left(-x-2\right)\left(x+9\right)$$

To find equation solutions, solve -x-2=0 and x+9=0.

$$x=-2 x=-9$$

Variable x cannot be equal to -2.

$$x=-9$$

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+2,2-x,x^{2}-4.

$$\left(x-2\right)\times 3x-\left(2+x\right)\left(2x+1\right)=2x^{2}+16$$

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

$$\left(3x-6\right)x-\left(2+x\right)\left(2x+1\right)=2x^{2}+16$$

Use the distributive property to multiply 3x-6 by x.

$$3x^{2}-6x-\left(2+x\right)\left(2x+1\right)=2x^{2}+16$$

Use the distributive property to multiply -1 by 2+x.

$$3x^{2}-6x+\left(-2-x\right)\left(2x+1\right)=2x^{2}+16$$

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

$$3x^{2}-6x-5x-2-2x^{2}=2x^{2}+16$$

Combine -6x and -5x to get -11x.

$$3x^{2}-11x-2-2x^{2}=2x^{2}+16$$

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

$$x^{2}-11x-2=2x^{2}+16$$

Subtract 2x^{2} from both sides.

$$x^{2}-11x-2-2x^{2}=16$$

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

$$-x^{2}-11x-2=16$$

Subtract 16 from both sides.

$$-x^{2}-11x-2-16=0$$

Subtract 16 from -2 to get -18.

$$-x^{2}-11x-18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -11 for b, and -18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\left(-1\right)\left(-18\right)}}{2\left(-1\right)}$$

Square -11.

$$x=\frac{-\left(-11\right)±\sqrt{121-4\left(-1\right)\left(-18\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-11\right)±\sqrt{121+4\left(-18\right)}}{2\left(-1\right)}$$

Multiply 4 times -18.

$$x=\frac{-\left(-11\right)±\sqrt{121-72}}{2\left(-1\right)}$$

Add 121 to -72.

$$x=\frac{-\left(-11\right)±\sqrt{49}}{2\left(-1\right)}$$

Take the square root of 49.

$$x=\frac{-\left(-11\right)±7}{2\left(-1\right)}$$

The opposite of -11 is 11.

$$x=\frac{11±7}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{11±7}{-2}$$

Now solve the equation x=\frac{11±7}{-2} when ± is plus. Add 11 to 7.

$$x=\frac{18}{-2}$$

Divide 18 by -2.

$$x=-9$$

Now solve the equation x=\frac{11±7}{-2} when ± is minus. Subtract 7 from 11.

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

Divide 4 by -2.

$$x=-2$$

The equation is now solved.

$$x=-9 x=-2$$

Variable x cannot be equal to -2.

$$x=-9$$

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+2,2-x,x^{2}-4.

$$\left(x-2\right)\times 3x-\left(2+x\right)\left(2x+1\right)=2x^{2}+16$$

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

$$\left(3x-6\right)x-\left(2+x\right)\left(2x+1\right)=2x^{2}+16$$

Use the distributive property to multiply 3x-6 by x.

$$3x^{2}-6x-\left(2+x\right)\left(2x+1\right)=2x^{2}+16$$

Use the distributive property to multiply -1 by 2+x.

$$3x^{2}-6x+\left(-2-x\right)\left(2x+1\right)=2x^{2}+16$$

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

$$3x^{2}-6x-5x-2-2x^{2}=2x^{2}+16$$

Combine -6x and -5x to get -11x.

$$3x^{2}-11x-2-2x^{2}=2x^{2}+16$$

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

$$x^{2}-11x-2=2x^{2}+16$$

Subtract 2x^{2} from both sides.

$$x^{2}-11x-2-2x^{2}=16$$

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

$$-x^{2}-11x-2=16$$

Add 2 to both sides.

$$-x^{2}-11x=16+2$$

Add 16 and 2 to get 18.

$$-x^{2}-11x=18$$

Divide both sides by -1.

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

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{11}{-1}\right)x=\frac{18}{-1}$$

Divide -11 by -1.

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

Divide 18 by -1.

$$x^{2}+11x=-18$$

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

$$x^{2}+11x+\left(\frac{11}{2}\right)^{2}=-18+\left(\frac{11}{2}\right)^{2}$$

Square \frac{11}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+11x+\frac{121}{4}=-18+\frac{121}{4}$$

Add -18 to \frac{121}{4}.

$$x^{2}+11x+\frac{121}{4}=\frac{49}{4}$$

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

$$\left(x+\frac{11}{2}\right)^{2}=\frac{49}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{11}{2}\right)^{2}}=\sqrt{\frac{49}{4}}$$

Simplify.

$$x+\frac{11}{2}=\frac{7}{2} x+\frac{11}{2}=-\frac{7}{2}$$

Subtract \frac{11}{2} from both sides of the equation.

$$x=-2 x=-9$$

Variable x cannot be equal to -2.

$$x=-9$$