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

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

Solve for x

$$x = -\frac{14}{11} = -1\frac{3}{11} \approx -1.272727273$$

$$x=1$$

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 3\left(x-2\right)\left(x+2\right), the least common multiple of x-2,x+2,3.

$$\left(3x+6\right)\times 3+\left(3x-6\right)x=\left(x^{2}-4\right)\left(-8\right)$$

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

$$9x+18+\left(3x-6\right)x=\left(x^{2}-4\right)\left(-8\right)$$

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

$$9x+18+3x^{2}-6x=\left(x^{2}-4\right)\left(-8\right)$$

Combine 9x and -6x to get 3x.

$$3x+18+3x^{2}=\left(x^{2}-4\right)\left(-8\right)$$

Use the distributive property to multiply x^{2}-4 by -8.

$$3x+18+3x^{2}=-8x^{2}+32$$

Add 8x^{2} to both sides.

$$3x+18+3x^{2}+8x^{2}=32$$

Combine 3x^{2} and 8x^{2} to get 11x^{2}.

$$3x+18+11x^{2}=32$$

Subtract 32 from both sides.

$$3x+18+11x^{2}-32=0$$

Subtract 32 from 18 to get -14.

$$3x-14+11x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$11x^{2}+3x-14=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 11 for a, 3 for b, and -14 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3±\sqrt{3^{2}-4\times 11\left(-14\right)}}{2\times 11}$$

Square 3.

$$x=\frac{-3±\sqrt{9-4\times 11\left(-14\right)}}{2\times 11}$$

Multiply -4 times 11.

$$x=\frac{-3±\sqrt{9-44\left(-14\right)}}{2\times 11}$$

Multiply -44 times -14.

$$x=\frac{-3±\sqrt{9+616}}{2\times 11}$$

Add 9 to 616.

$$x=\frac{-3±\sqrt{625}}{2\times 11}$$

Take the square root of 625.

$$x=\frac{-3±25}{2\times 11}$$

Multiply 2 times 11.

$$x=\frac{-3±25}{22}$$

Now solve the equation x=\frac{-3±25}{22} when ± is plus. Add -3 to 25.

$$x=\frac{22}{22}$$

Divide 22 by 22.

$$x=1$$

Now solve the equation x=\frac{-3±25}{22} when ± is minus. Subtract 25 from -3.

$$x=-\frac{28}{22}$$

Reduce the fraction \frac{-28}{22} to lowest terms by extracting and canceling out 2.

$$x=-\frac{14}{11}$$

The equation is now solved.

$$x=1 x=-\frac{14}{11}$$

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 3\left(x-2\right)\left(x+2\right), the least common multiple of x-2,x+2,3.

$$\left(3x+6\right)\times 3+\left(3x-6\right)x=\left(x^{2}-4\right)\left(-8\right)$$

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

$$9x+18+\left(3x-6\right)x=\left(x^{2}-4\right)\left(-8\right)$$

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

$$9x+18+3x^{2}-6x=\left(x^{2}-4\right)\left(-8\right)$$

Combine 9x and -6x to get 3x.

$$3x+18+3x^{2}=\left(x^{2}-4\right)\left(-8\right)$$

Use the distributive property to multiply x^{2}-4 by -8.

$$3x+18+3x^{2}=-8x^{2}+32$$

Add 8x^{2} to both sides.

$$3x+18+3x^{2}+8x^{2}=32$$

Combine 3x^{2} and 8x^{2} to get 11x^{2}.

$$3x+18+11x^{2}=32$$

Subtract 18 from both sides.

$$3x+11x^{2}=32-18$$

Subtract 18 from 32 to get 14.

$$3x+11x^{2}=14$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$11x^{2}+3x=14$$

Divide both sides by 11.

$$\frac{11x^{2}+3x}{11}=\frac{14}{11}$$

Dividing by 11 undoes the multiplication by 11.

$$x^{2}+\frac{3}{11}x=\frac{14}{11}$$

Divide \frac{3}{11}, the coefficient of the x term, by 2 to get \frac{3}{22}. Then add the square of \frac{3}{22} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{3}{11}x+\left(\frac{3}{22}\right)^{2}=\frac{14}{11}+\left(\frac{3}{22}\right)^{2}$$

Square \frac{3}{22} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{3}{11}x+\frac{9}{484}=\frac{14}{11}+\frac{9}{484}$$

Add \frac{14}{11} to \frac{9}{484} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{3}{11}x+\frac{9}{484}=\frac{625}{484}$$

Factor x^{2}+\frac{3}{11}x+\frac{9}{484}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{3}{22}\right)^{2}=\frac{625}{484}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{22}\right)^{2}}=\sqrt{\frac{625}{484}}$$

Simplify.

$$x+\frac{3}{22}=\frac{25}{22} x+\frac{3}{22}=-\frac{25}{22}$$

Subtract \frac{3}{22} from both sides of the equation.

$$x=1 x=-\frac{14}{11}$$