If `alpha and beta` are the roots of the equation `ax^(2) + bx + b = 0`. Then what is the value of `sqrt((alpha)/(beta))+sqrt((beta)/(alpha))+sqrt((b)/(a))=?`

Options

A. -10
B. 0
C. 1
D. 2

Answer

To solve the problem, we need to find the value of the expression:

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

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

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

### Step 1: Identify the roots and their properties
From Vieta's formulas, we know that for a quadratic equation \(ax^2 + bx + c = 0\):
- The sum of the roots \(\alpha + \beta = -\frac{b}{a}\)
- The product of the roots \(\alpha \beta = \frac{c}{a}\)

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

### Step 2: Calculate \(\sqrt{\frac{\alpha}{\beta}} + \sqrt{\frac{\beta}{\alpha}}\)
We can simplify this expression as follows:

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

### Step 3: Substitute the values from Vieta's formulas
Substituting the values we found from Vieta's formulas:

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

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

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

### Step 5: Add \(\sqrt{\frac{b}{a}}\) to the expression
Now we need to add \(\sqrt{\frac{b}{a}}\):

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

### Step 6: Combine the fractions
To combine these fractions, we find a common denominator:

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

### Step 7: Conclusion
Thus, the final expression simplifies to:

\[ \frac{b - a}{\sqrt{ab}} \]

### Final Answer
The value of \(\sqrt{\frac{\alpha}{\beta}} + \sqrt{\frac{\beta}{\alpha}} + \sqrt{\frac{b}{a}} = \frac{b - a}{\sqrt{ab}}\).

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