80 /( 18 + x) + 80 / (18 - x) - 9 = 0

To solve the equation

\[ \frac{80}{18 + x} + \frac{80}{18 - x} - 9 = 0, \]

let's first combine the fractions. Starting with the two fractions:

\[ \frac{80}{18 + x} + \frac{80}{18 - x}, \]

we can find a common denominator, which is \((18 + x)(18 - x)\):

\[ \frac{80(18 - x) + 80(18 + x)}{(18 + x)(18 - x)}. \]

Now, simplifying the numerator:

\[ 80(18 - x) + 80(18 + x) = 80 \cdot 18 - 80x + 80 \cdot 18 + 80x = 2 \cdot 80 \cdot 18 = 1440. \]

So we can rewrite the equation:

\[ \frac{1440}{(18 + x)(18 - x)} - 9 = 0. \]

Now, let’s isolate the fraction:

\[ \frac{1440}{(18 + x)(18 - x)} = 9. \]

Next, we can cross-multiply:

\[ 1440 = 9(18 + x)(18 - x). \]

Expanding the right side using the difference of squares:

\[ 1440 = 9(324 - x^2). \]

Distributing the 9:

\[ 1440 = 2916 - 9x^2. \]

Rearranging to bring all terms to one side:

\[ 9x^2 + 1440 - 2916 = 0, \] \[ 9x^2 - 1476 = 0. \]

Dividing the entire equation by 9:

\[ x^2 - 164 = 0. \]

Now, we can solve for \(x^2\):

\[ x^2 = 164. \]

Taking the square root of both sides gives us:

\[ x = \pm \sqrt{164} = \pm 2\sqrt{41}. \]

Thus, the solutions to the equation are:

\[ x = 2\sqrt{41} \quad \text{and} \quad x = -2\sqrt{41}. \]