If `alpha` and `beta` are the zeros of the quadratic polynomial `f(x)=6x^2+x-2` , find the value of `alpha/beta+beta/alpha` .

Options

A. ` (25)/12 `
B. ` (-25)/11 `
C. ` (-25)/12 `
D. none of these

Answer

To find the value of \(\frac{\alpha}{\beta} + \frac{\beta}{\alpha}\) where \(\alpha\) and \(\beta\) are the zeros of the quadratic polynomial \(f(x) = 6x^2 + x - 2\), we can follow these steps:

### Step 1: Identify coefficients
The given polynomial is \(f(x) = 6x^2 + x - 2\). Here, we have:
- \(a = 6\)
- \(b = 1\)
- \(c = -2\)

### Step 2: Calculate the sum and product of the roots
Using Vieta's formulas:
- The sum of the roots \(\alpha + \beta = -\frac{b}{a} = -\frac{1}{6}\)
- The product of the roots \(\alpha \beta = \frac{c}{a} = \frac{-2}{6} = -\frac{1}{3}\)

### Step 3: Use the identity for \(\frac{\alpha}{\beta} + \frac{\beta}{\alpha}\)
We know that:
\[ \frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{\alpha^2 + \beta^2}{\alpha \beta} \]
To find \(\alpha^2 + \beta^2\), we can use the identity:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \]

### Step 4: Calculate \((\alpha + \beta)^2\)
Substituting the values we found:
\[ (\alpha + \beta)^2 = \left(-\frac{1}{6}\right)^2 = \frac{1}{36} \]

### Step 5: Calculate \(2\alpha\beta\)
Now, calculate \(2\alpha\beta\):
\[ 2\alpha\beta = 2 \times -\frac{1}{3} = -\frac{2}{3} \]

### Step 6: Substitute into the identity
Now substituting back into the identity:
\[ \alpha^2 + \beta^2 = \frac{1}{36} - \left(-\frac{2}{3}\right) \]
To combine these, we need a common denominator:
\[ -\frac{2}{3} = -\frac{2 \times 12}{3 \times 12} = -\frac{24}{36} \]
Thus:
\[ \alpha^2 + \beta^2 = \frac{1}{36} + \frac{24}{36} = \frac{25}{36} \]

### Step 7: Substitute into \(\frac{\alpha^2 + \beta^2}{\alpha \beta}\)
Now we can substitute \(\alpha^2 + \beta^2\) and \(\alpha \beta\) into our equation:
\[ \frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{\frac{25}{36}}{-\frac{1}{3}} = \frac{25}{36} \times -3 = -\frac{25 \times 3}{36} = -\frac{75}{36} \]
Simplifying \(-\frac{75}{36}\):
\[ -\frac{75}{36} = -\frac{25}{12} \]

### Final Answer
Thus, the value of \(\frac{\alpha}{\beta} + \frac{\beta}{\alpha}\) is \(-\frac{25}{12}\).

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