The roots of the equation `3 sqrtx+ 5(x)^(-(1)/(2))= sqrt2` can be found by solving____

Options

A. `9x^(2) + 28x+ 25= 0`
B. `9x^(2) + 30x+ 25= 0`
C. `9x^(2) + 28x- 25= 0`
D. `16x^(2) + 22x- 30 = 0`

Answer

To solve the equation \( 3 \sqrt{x} + 5 x^{-\frac{1}{2}} = \sqrt{2} \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ 3 \sqrt{x} + 5 x^{-\frac{1}{2}} = \sqrt{2} \]
We can rewrite \( x^{-\frac{1}{2}} \) as \( \frac{1}{\sqrt{x}} \):
\[ 3 \sqrt{x} + \frac{5}{\sqrt{x}} = \sqrt{2} \]

### Step 2: Multiply through by \(\sqrt{x}\)
To eliminate the fraction, we multiply every term by \(\sqrt{x}\):
\[ 3x + 5 = \sqrt{2} \sqrt{x} \]

### Step 3: Isolate the square root
Rearranging gives us:
\[ \sqrt{2} \sqrt{x} = 3x + 5 \]

### Step 4: Square both sides
Next, we square both sides to eliminate the square root:
\[ (\sqrt{2} \sqrt{x})^2 = (3x + 5)^2 \]
This simplifies to:
\[ 2x = 9x^2 + 30x + 25 \]

### Step 5: Rearrange into standard quadratic form
Rearranging gives us:
\[ 9x^2 + 30x + 25 - 2x = 0 \]
This simplifies to:
\[ 9x^2 + 28x + 25 = 0 \]

### Conclusion
The roots of the equation can be found by solving the quadratic equation:
\[ 9x^2 + 28x + 25 = 0 \]