Evaluation Using Zeros of a Quadratic Polynomial

Video Explanation

Question

If \( \alpha \) and \( \beta \) are the zeroes of the polynomial

\[ p(x) = 4x^2 + 3x + 7, \]

find

\[ \frac{1}{\alpha} + \frac{1}{\beta}. \]

Solution

Step 1: Write Relations Between Zeroes and Coefficients

For the quadratic polynomial \(4x^2 + 3x + 7\),

\[ \alpha + \beta = -\frac{b}{a} = -\frac{3}{4}, \quad \alpha\beta = \frac{c}{a} = \frac{7}{4} \]

Step 2: Evaluate the Required Expression

\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha\beta} \]

\[ = \frac{-\frac{3}{4}}{\frac{7}{4}} = -\frac{3}{7} \]

Conclusion

\[ \boxed{\frac{1}{\alpha} + \frac{1}{\beta} = -\frac{3}{7}} \]