If `sqrt(x)+(58)/(sqrt(x))` =31 then which of the following can be the value of x?

Options

A. 9
B. 1
C. 2
D. 4

Explanation

To solve the equation \( \sqrt{x} + \frac{58}{\sqrt{x}} = 31 \), we can follow these steps:

### Step 1: Substitute \( \sqrt{x} \) with \( t \)
Let \( t = \sqrt{x} \). Then the equation becomes:
\[ t + \frac{58}{t} = 31 \]

### Step 2: Multiply through by \( t \) to eliminate the fraction
Multiply both sides by \( t \) (assuming \( t \neq 0 \)):
\[ t^2 + 58 = 31t \]

### Step 3: Rearrange the equation
Rearranging gives us:
\[ t^2 - 31t + 58 = 0 \]

### Step 4: Factor the quadratic equation
We need to factor the quadratic equation \( t^2 - 31t + 58 = 0 \). We look for two numbers that multiply to \( 58 \) and add to \( -31 \). The numbers are \( -29 \) and \( -2 \):
\[ (t - 29)(t - 2) = 0 \]

### Step 5: Solve for \( t \)
Setting each factor to zero gives us:
\[ t - 29 = 0 \quad \text{or} \quad t - 2 = 0 \]
Thus, we have:
\[ t = 29 \quad \text{or} \quad t = 2 \]

### Step 6: Substitute back to find \( x \)
Recall that \( t = \sqrt{x} \):
1. If \( t = 29 \):
\[ \sqrt{x} = 29 \implies x = 29^2 = 841 \]
2. If \( t = 2 \):
\[ \sqrt{x} = 2 \implies x = 2^2 = 4 \]

### Conclusion
The possible values of \( x \) are \( 841 \) and \( 4 \). Since the question asks for which of the following can be the value of \( x \), we check the options provided. If \( 4 \) is one of the options, then it is a valid answer.