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

Step 1: Understanding the Concept:
To convert a fractional equation into a quadratic, we find a common denominator and multiply the entire equation by it to clear the fractions.
Step 2: Key Formula or Approach:
1. Common Denominator: $(18-x)(18+x) = 324 - x^2$.
2. Discriminant $D = b^2 - 4ac$.
3. Quadratic Formula: $x = \frac{-b \pm \sqrt{D}}{2a}$.
Step 3: Detailed Explanation:
1. Simplify the equation:
\[ 24 \left( \frac{1}{18-x} - \frac{1}{18+x} \right) = 1 \]
\[ 24 \left( \frac{(18+x) - (18-x)}{324 - x^2} \right) = 1 \]
\[ 24 \left( \frac{2x}{324 - x^2} \right) = 1 \implies 48x = 324 - x^2 \] 2. Standard Form: $x^2 + 48x - 324 = 0$.
3. Discriminant ($D$):
- $a=1, b=48, c=-324$.
- $D = (48)^2 - 4(1)(-324) = 2304 + 1296 = 3600$.
4. Roots:
- $x = \frac{-48 \pm \sqrt{3600}}{2(1)} = \frac{-48 \pm 60}{2}$.
- $x_1 = \frac{12}{2} = 6$; $x_2 = \frac{-108}{2} = -54$.
Step 4: Final Answer:
Standard form: $x^2 + 48x - 324 = 0$. Discriminant: 3600. Roots: 6 and -54.