Question

Express $\frac{24}{18-x} - \frac{24}{18+x} = 1$ as a quadratic equation in standard form and find the discriminant. Also, find the roots of the equation.

Hint:

When a problem specifies "positive numbers," always reject the negative root obtained from the quadratic equation.

Answer 1

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