Concept Applied:
To solve a fractional equation, we first remove the denominators by taking the LCM. This converts the equation into a quadratic form, which can then be solved using the quadratic formula.
Stepwise Simplification:
Given equation:
\[
24\left(\frac{1}{18 - x} - \frac{1}{18 + x}\right) = 1
\]
1) Take common denominator:
$(18 - x)(18 + x) = 324 - x^2$
2) Combine the fractions:
\[
24 \left( \frac{(18 + x) - (18 - x)}{324 - x^2} \right) = 1
\]
3) Simplify numerator:
\[
(18 + x) - (18 - x) = 2x
\]
So,
\[
24 \left( \frac{2x}{324 - x^2} \right) = 1
\]
4) Multiply:
\[
\frac{48x}{324 - x^2} = 1
\]
5) Cross multiply:
\[
48x = 324 - x^2
\]
6) Rearrange in standard quadratic form:
\[
x^2 + 48x - 324 = 0
\]
Finding Discriminant:
$a = 1, \; b = 48, \; c = -324$
\[
D = b^2 - 4ac
\]
\[
D = 48^2 - 4(1)(-324)
\]
\[
D = 2304 + 1296 = 3600
\]
Finding Roots:
\[
x = \frac{-b \pm \sqrt{D}}{2a}
\]
\[
x = \frac{-48 \pm \sqrt{3600}}{2}
\]
\[
x = \frac{-48 \pm 60}{2}
\]
Therefore,
\[
x = 6 \quad \text{or} \quad x = -54
\]
Final Result:
Quadratic equation: $x^2 + 48x - 324 = 0$
Discriminant = 3600
Roots = 6 and -54