If `alpha` and `beta` are the roots of the equation `x^2+sqrt(alpha)x+beta=0` then the values of `alpha` and `beta` are -

Options

A. `alpha = 1, beta = -1`
B. `alpha = 1, beta = -2`
C. `alpha = 2, beta = 1`
D. `alpha = 2, beta = -2`

Answer

To solve the problem, we need to find the values of \( \alpha \) and \( \beta \) given that they are the roots of the equation:

\[ x^2 + \sqrt{\alpha} x + \beta = 0 \]

### Step 1: Use the relationships for roots of a quadratic equation
For a quadratic equation of the form \( ax^2 + bx + c = 0 \), the sum and product of the roots can be expressed as:
- Sum of the roots \( \alpha + \beta = -\frac{b}{a} \)
- Product of the roots \( \alpha \beta = \frac{c}{a} \)

In our case, \( a = 1 \), \( b = \sqrt{\alpha} \), and \( c = \beta \). Thus, we can write:

\[ \alpha + \beta = -\sqrt{\alpha} \quad \text{(Equation 1)} \]
\[ \alpha \beta = \beta \quad \text{(Equation 2)} \]

### Step 2: Analyze Equation 2
From Equation 2, we have:

\[ \alpha \beta = \beta \]

Assuming \( \beta \neq 0 \), we can divide both sides by \( \beta \):

\[ \alpha = 1 \]

### Step 3: Substitute \( \alpha \) back into Equation 1
Now that we have \( \alpha = 1 \), we can substitute this value into Equation 1:

\[ 1 + \beta = -\sqrt{1} \]

Since \( \sqrt{1} = 1 \), we can simplify this to:

\[ 1 + \beta = -1 \]

### Step 4: Solve for \( \beta \)
To find \( \beta \), we rearrange the equation:

\[ \beta = -1 - 1 \]
\[ \beta = -2 \]

### Conclusion
Thus, the values of \( \alpha \) and \( \beta \) are:

\[ \alpha = 1 \quad \text{and} \quad \beta = -2 \]

### Final Answer
The values of \( \alpha \) and \( \beta \) are \( \alpha = 1 \) and \( \beta = -2 \).

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