If `alpha, beta` are the zeros of the polynomial `4x ^(2)+3x+7` then find the value of `(1)/(alpha) + (1)/(beta).`

Answer

To find the value of \(\frac{1}{\alpha} + \frac{1}{\beta}\) where \(\alpha\) and \(\beta\) are the zeros of the polynomial \(4x^2 + 3x + 7\), we can follow these steps:

### Step 1: Identify coefficients
The polynomial is in the standard form \(ax^2 + bx + c\). Here, we have:
- \(a = 4\)
- \(b = 3\)
- \(c = 7\)

### Step 2: Use Vieta's formulas
According to Vieta's formulas:
- The sum of the roots \(\alpha + \beta = -\frac{b}{a}\)
- The product of the roots \(\alpha \beta = \frac{c}{a}\)

### Step 3: Calculate \(\alpha + \beta\)
Using the values of \(b\) and \(a\):
\[ \alpha + \beta = -\frac{3}{4} \]

### Step 4: Calculate \(\alpha \beta\)
Using the values of \(c\) and \(a\):
\[ \alpha \beta = \frac{7}{4} \]

### Step 5: Find \(\frac{1}{\alpha} + \frac{1}{\beta}\)
We can express \(\frac{1}{\alpha} + \frac{1}{\beta}\) in terms of the sum and product of the roots:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\beta + \alpha}{\alpha \beta} \]

### Step 6: Substitute the values
Substituting the values we found:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha \beta} = \frac{-\frac{3}{4}}{\frac{7}{4}} \]

### Step 7: Simplify the expression
When we simplify this, we get:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{-3}{4} \times \frac{4}{7} = -\frac{3}{7} \]

### Final Answer
Thus, the value of \(\frac{1}{\alpha} + \frac{1}{\beta}\) is:
\[ -\frac{3}{7} \]

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