Solve x/x+4+x/x-4=54/9

$$\frac{ x }{ x+4 } + \frac{ x }{ x-4 } = 5 \frac{ 4 }{ 9 }$$

Solve for x

$$x = \frac{28 \sqrt{31}}{31} \approx 5.028948457$$

$$x = -\frac{28 \sqrt{31}}{31} \approx -5.028948457$$

Steps by Finding Square Root

Variable x cannot be equal to any of the values -4,4 since division by zero is not defined. Multiply both sides of the equation by 9\left(x-4\right)\left(x+4\right), the least common multiple of x+4,x-4,9.

$$\left(9x-36\right)x+\left(9x+36\right)x=\left(x^{2}-16\right)\left(5\times 9+4\right)$$

Use the distributive property to multiply 9x-36 by x.

$$9x^{2}-36x+\left(9x+36\right)x=\left(x^{2}-16\right)\left(5\times 9+4\right)$$

Use the distributive property to multiply 9x+36 by x.

$$9x^{2}-36x+9x^{2}+36x=\left(x^{2}-16\right)\left(5\times 9+4\right)$$

Combine 9x^{2} and 9x^{2} to get 18x^{2}.

$$18x^{2}-36x+36x=\left(x^{2}-16\right)\left(5\times 9+4\right)$$

Combine -36x and 36x to get 0.

$$18x^{2}=\left(x^{2}-16\right)\left(5\times 9+4\right)$$

Multiply 5 and 9 to get 45.

$$18x^{2}=\left(x^{2}-16\right)\left(45+4\right)$$

Add 45 and 4 to get 49.

$$18x^{2}=\left(x^{2}-16\right)\times 49$$

Use the distributive property to multiply x^{2}-16 by 49.

$$18x^{2}=49x^{2}-784$$

Subtract 49x^{2} from both sides.

$$18x^{2}-49x^{2}=-784$$

Combine 18x^{2} and -49x^{2} to get -31x^{2}.

$$-31x^{2}=-784$$

Divide both sides by -31.

$$x^{2}=\frac{-784}{-31}$$

Fraction \frac{-784}{-31} can be simplified to \frac{784}{31} by removing the negative sign from both the numerator and the denominator.

$$x^{2}=\frac{784}{31}$$

Take the square root of both sides of the equation.

$$x=\frac{28\sqrt{31}}{31} x=-\frac{28\sqrt{31}}{31}$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -4,4 since division by zero is not defined. Multiply both sides of the equation by 9\left(x-4\right)\left(x+4\right), the least common multiple of x+4,x-4,9.

$$\left(9x-36\right)x+\left(9x+36\right)x=\left(x^{2}-16\right)\left(5\times 9+4\right)$$

Use the distributive property to multiply 9x-36 by x.

$$9x^{2}-36x+\left(9x+36\right)x=\left(x^{2}-16\right)\left(5\times 9+4\right)$$

Use the distributive property to multiply 9x+36 by x.

$$9x^{2}-36x+9x^{2}+36x=\left(x^{2}-16\right)\left(5\times 9+4\right)$$

Combine 9x^{2} and 9x^{2} to get 18x^{2}.

$$18x^{2}-36x+36x=\left(x^{2}-16\right)\left(5\times 9+4\right)$$

Combine -36x and 36x to get 0.

$$18x^{2}=\left(x^{2}-16\right)\left(5\times 9+4\right)$$

Multiply 5 and 9 to get 45.

$$18x^{2}=\left(x^{2}-16\right)\left(45+4\right)$$

Add 45 and 4 to get 49.

$$18x^{2}=\left(x^{2}-16\right)\times 49$$

Use the distributive property to multiply x^{2}-16 by 49.

$$18x^{2}=49x^{2}-784$$

Subtract 49x^{2} from both sides.

$$18x^{2}-49x^{2}=-784$$

Combine 18x^{2} and -49x^{2} to get -31x^{2}.

$$-31x^{2}=-784$$

Add 784 to both sides.

$$-31x^{2}+784=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -31 for a, 0 for b, and 784 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-31\right)\times 784}}{2\left(-31\right)}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-31\right)\times 784}}{2\left(-31\right)}$$

Multiply -4 times -31.

$$x=\frac{0±\sqrt{124\times 784}}{2\left(-31\right)}$$

Multiply 124 times 784.

$$x=\frac{0±\sqrt{97216}}{2\left(-31\right)}$$

Take the square root of 97216.

$$x=\frac{0±56\sqrt{31}}{2\left(-31\right)}$$

Multiply 2 times -31.

$$x=\frac{0±56\sqrt{31}}{-62}$$

Now solve the equation x=\frac{0±56\sqrt{31}}{-62} when ± is plus.

$$x=-\frac{28\sqrt{31}}{31}$$

Now solve the equation x=\frac{0±56\sqrt{31}}{-62} when ± is minus.

$$x=\frac{28\sqrt{31}}{31}$$

The equation is now solved.

$$x=-\frac{28\sqrt{31}}{31} x=\frac{28\sqrt{31}}{31}$$