Evaluation of an Expression Using Zeros of a Quadratic Polynomial
Video Explanation
Question
If \( \alpha \) and \( \beta \) are the zeroes of the quadratic polynomial
\[ f(x) = ax^2 + bx + c, \]
evaluate
\[ \frac{1}{\alpha} + \frac{1}{\beta} – 2\alpha\beta. \]
Solution
Step 1: Write Relations Between Zeros and Coefficients
For the quadratic polynomial \( ax^2 + bx + c \),
\[ \alpha + \beta = -\frac{b}{a}, \quad \alpha\beta = \frac{c}{a} \]
Step 2: Find \( \dfrac{1}{\alpha} + \dfrac{1}{\beta} \)
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha\beta} \]
\[ = \frac{-\dfrac{b}{a}}{\dfrac{c}{a}} = -\frac{b}{c} \]
Step 3: Evaluate the Given Expression
\[ \frac{1}{\alpha} + \frac{1}{\beta} – 2\alpha\beta \]
\[ = -\frac{b}{c} – 2\left(\frac{c}{a}\right) \]
\[ = -\frac{b}{c} – \frac{2c}{a} \]
Conclusion
The required value is:
\[ \boxed{-\frac{b}{c} – \frac{2c}{a}} \]
\[ \therefore \quad \frac{1}{\alpha} + \frac{1}{\beta} – 2\alpha\beta = -\frac{b}{c} – \frac{2c}{a}. \]