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}.
]
