Find x if `(sqrt(5x) + sqrt(3x + 1))/(sqrt(5x) -sqrt(3x + 1))` = 9 .

Options

A. 5
B. 9
C. 4
D. 1

Answer

To solve the equation \(\frac{\sqrt{5x} + \sqrt{3x + 1}}{\sqrt{5x} - \sqrt{3x + 1}} = 9\), we will follow these steps:

### Step 1: Apply the Compando and Dividendo Rule
We can apply the compando and dividendo rule which states that if \(\frac{a}{b} = \frac{c}{d}\), then:
\[ \frac{a + b}{a - b} = \frac{c + d}{c - d} \]
In our case, let:
- \(a = \sqrt{5x}\)
- \(b = \sqrt{3x + 1}\)
- \(c = 9\)
- \(d = 1\)

Thus, we can rewrite the equation as:
\[ \frac{\sqrt{5x} + \sqrt{3x + 1} + \sqrt{5x} - \sqrt{3x + 1}}{\sqrt{5x} + \sqrt{3x + 1} - \sqrt{5x} + \sqrt{3x + 1}} = \frac{9 + 1}{9 - 1} \]

### Step 2: Simplify the Equation
The numerator simplifies to:
\[ 2\sqrt{5x} \]
And the denominator simplifies to:
\[ 2\sqrt{3x + 1} \]
Thus, we have:
\[ \frac{2\sqrt{5x}}{2\sqrt{3x + 1}} = \frac{10}{8} \]
This simplifies to:
\[ \frac{\sqrt{5x}}{\sqrt{3x + 1}} = \frac{5}{4} \]

### Step 3: Cross Multiply
Cross multiplying gives us:
\[ 4\sqrt{5x} = 5\sqrt{3x + 1} \]

### Step 4: Square Both Sides
Squaring both sides to eliminate the square roots:
\[ (4\sqrt{5x})^2 = (5\sqrt{3x + 1})^2 \]
This simplifies to:
\[ 16 \cdot 5x = 25(3x + 1) \]
\[ 80x = 75x + 25 \]

### Step 5: Solve for x
Rearranging gives:
\[ 80x - 75x = 25 \]
\[ 5x = 25 \]
Dividing both sides by 5:
\[ x = 5 \]

### Final Answer
Thus, the value of \(x\) is:
\[ \boxed{5} \]

---