If `(3x)/5+5/(2x)=4`, then find `(9x^(2))/25 + 25/(4x^(2))`

Answer

To solve the equation \(\frac{3x}{5} + \frac{5}{2x} = 4\) and find the value of \(\frac{9x^2}{25} + \frac{25}{4x^2}\), we can follow these steps:

### Step 1: Square both sides of the equation
We start with the equation:
\[ \frac{3x}{5} + \frac{5}{2x} = 4 \]
Now, we square both sides:
\[ \left(\frac{3x}{5} + \frac{5}{2x}\right)^2 = 4^2 \]
This gives us:
\[ \left(\frac{3x}{5} + \frac{5}{2x}\right)^2 = 16 \]

### Step 2: Expand the left side using the formula for \((a + b)^2\)
Using the formula \((a + b)^2 = a^2 + b^2 + 2ab\), we can expand the left side:
\[ \left(\frac{3x}{5}\right)^2 + \left(\frac{5}{2x}\right)^2 + 2\left(\frac{3x}{5}\right)\left(\frac{5}{2x}\right) = 16 \]

### Step 3: Calculate each term
Calculating each term:
1. \(\left(\frac{3x}{5}\right)^2 = \frac{9x^2}{25}\)
2. \(\left(\frac{5}{2x}\right)^2 = \frac{25}{4x^2}\)
3. \(2\left(\frac{3x}{5}\right)\left(\frac{5}{2x}\right) = 2 \cdot \frac{3x \cdot 5}{5 \cdot 2x} = 2 \cdot \frac{3}{2} = 3\)

Putting it all together, we have:
\[ \frac{9x^2}{25} + \frac{25}{4x^2} + 3 = 16 \]

### Step 4: Isolate the expression we want to find
Now, we can isolate \(\frac{9x^2}{25} + \frac{25}{4x^2}\):
\[ \frac{9x^2}{25} + \frac{25}{4x^2} = 16 - 3 \]
This simplifies to:
\[ \frac{9x^2}{25} + \frac{25}{4x^2} = 13 \]

### Final Answer
Thus, the value of \(\frac{9x^2}{25} + \frac{25}{4x^2}\) is:
\[ \boxed{13} \]