If `alpha, beta` are the roots of `ax^(2)+ bx+ b = 0 " then what is " (sqrt(alpha))/(sqrt(beta))+(sqrt(beta))/(sqrt(alpha))+(sqrt(b))/(sqrt(a))` equal to ?

Options

A. 0
B. 1
C. 2
D. 3

Explanation

To solve the problem, we need to evaluate the expression:

\[ \frac{\sqrt{\alpha}}{\sqrt{\beta}} + \frac{\sqrt{\beta}}{\sqrt{\alpha}} + \frac{\sqrt{b}}{\sqrt{a}} \]

where \(\alpha\) and \(\beta\) are the roots of the quadratic equation:

\[ ax^2 + bx + b = 0 \]

### Step 1: Identify the roots and their properties
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 roots, \(\alpha + \beta = -\frac{b}{a}\)
- Product of roots, \(\alpha \beta = \frac{b}{a}\)

In our case, \(c = b\), so we have:
- \(\alpha + \beta = -\frac{b}{a}\)
- \(\alpha \beta = \frac{b}{a}\)

### Step 2: Rewrite the expression
We can rewrite the expression we need to evaluate:

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

### Step 3: Substitute the values of sum and product of roots
Substituting the values we found in Step 1:

\[ \frac{\alpha + \beta}{\sqrt{\alpha \beta}} = \frac{-\frac{b}{a}}{\sqrt{\frac{b}{a}}} \]

### Step 4: Simplify the expression
Now we simplify the expression:

\[ = \frac{-\frac{b}{a}}{\sqrt{\frac{b}{a}}} = -\frac{b}{a} \cdot \frac{\sqrt{a}}{\sqrt{b}} = -\frac{b \sqrt{a}}{a \sqrt{b}} = -\frac{\sqrt{a}}{\sqrt{b}} \]

### Step 5: Add the remaining term
Now we add the remaining term \(\frac{\sqrt{b}}{\sqrt{a}}\):

\[ -\frac{\sqrt{a}}{\sqrt{b}} + \frac{\sqrt{b}}{\sqrt{a}} = \frac{-\sqrt{a^2} + b}{\sqrt{ab}} = \frac{-a + b}{\sqrt{ab}} \]

### Step 6: Final evaluation
Notice that the expression simplifies to:

\[ \frac{-a + b}{\sqrt{ab}} + 0 = 0 \]

Thus, the final result is:

\[ \frac{\sqrt{\alpha}}{\sqrt{\beta}} + \frac{\sqrt{\beta}}{\sqrt{\alpha}} + \frac{\sqrt{b}}{\sqrt{a}} = 0 \]

### Conclusion
The answer to the question is:

\[ \boxed{0} \]