Step 1: Understanding the Concept:
The sum of the reciprocals of the roots of a quadratic equation can be expressed in terms of the sum and product of the roots.
: Key Formula or Approach:
For \( ax^{2} + bx + c = 0 \):
Sum of roots \( (\alpha + \beta) = -b/a \).
Product of roots \( (\alpha\beta) = c/a \).
Value required: \( \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha\beta} \).
Step 2: Detailed Explanation:
From the equation \( 4x^{2} + 3x + 7 = 0 \):
\( a = 4, b = 3, c = 7 \).
Sum of roots \( \alpha + \beta = -3/4 \).
Product of roots \( \alpha\beta = 7/4 \).
Substituting these into the expression:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{-3/4}{7/4} \]
The denominators (4) cancel out:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = -\frac{3}{7} \].
Step 3: Final Answer:
The value is \( -3/7 \).