The positive value of x in the equation `(24)/(18-x)-(24)/(18+x)=1` is :

Options

A. 5
B. 6
C. 9
D. 11

Answer

To solve the equation \(\frac{24}{18-x} - \frac{24}{18+x} = 1\), we will follow these steps:

### Step 1: Find a common denominator
The common denominator for the fractions \(\frac{24}{18-x}\) and \(\frac{24}{18+x}\) is \((18-x)(18+x)\).

### Step 2: Rewrite the equation
Rewriting the equation with the common denominator, we have:
\[ \frac{24(18+x) - 24(18-x)}{(18-x)(18+x)} = 1 \]

### Step 3: Simplify the numerator
Now, simplify the numerator:
\[ 24(18+x) - 24(18-x) = 24 \cdot 18 + 24x - 24 \cdot 18 + 24x = 48x \]
Thus, the equation becomes:
\[ \frac{48x}{(18-x)(18+x)} = 1 \]

### Step 4: Cross-multiply
Cross-multiplying gives:
\[ 48x = (18-x)(18+x) \]

### Step 5: Expand the right side
Expanding the right side:
\[ (18-x)(18+x) = 18^2 - x^2 = 324 - x^2 \]
So, we have:
\[ 48x = 324 - x^2 \]

### Step 6: Rearrange the equation
Rearranging gives:
\[ x^2 + 48x - 324 = 0 \]

### Step 7: Use the quadratic formula
To solve for \(x\), we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \(a = 1\), \(b = 48\), and \(c = -324\).

### Step 8: Calculate the discriminant
Calculating the discriminant:
\[ b^2 - 4ac = 48^2 - 4 \cdot 1 \cdot (-324) = 2304 + 1296 = 3600 \]

### Step 9: Solve for \(x\)
Now substituting back into the quadratic formula:
\[ x = \frac{-48 \pm \sqrt{3600}}{2 \cdot 1} = \frac{-48 \pm 60}{2} \]
Calculating the two possible values:
1. \(x = \frac{12}{2} = 6\)
2. \(x = \frac{-108}{2} = -54\) (not a positive value)

### Conclusion
The positive value of \(x\) is \(6\).