Evaluation of \( \alpha^2\beta – \alpha\beta^2 \)

Video Explanation

Question

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

\[ f(x) = ax^2 + bx + c, \]

evaluate

\[ \alpha^2\beta – \alpha\beta^2. \]

Solution

Step 1: Use 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: Simplify the Given Expression

\[ \alpha^2\beta – \alpha\beta^2 \]

Take \( \alpha\beta \) common:

\[ = \alpha\beta(\alpha – \beta) \]

Step 3: Find \( \alpha – \beta \)

\[ (\alpha – \beta)^2 = (\alpha + \beta)^2 – 4\alpha\beta \]

\[ = \left(-\frac{b}{a}\right)^2 – 4\left(\frac{c}{a}\right) = \frac{b^2 – 4ac}{a^2} \]

\[ \Rightarrow \alpha – \beta = \frac{\sqrt{b^2 – 4ac}}{a} \]

Step 4: Substitute the Values

\[ \alpha^2\beta – \alpha\beta^2 = \left(\frac{c}{a}\right)\left(\frac{\sqrt{b^2 – 4ac}}{a}\right) \]

\[ = \frac{c\sqrt{b^2 – 4ac}}{a^2} \]

Conclusion

The required value is:

\[ \boxed{\alpha^2\beta – \alpha\beta^2 = \frac{c\sqrt{b^2 – 4ac}}{a^2}} \]

\[ \therefore \quad \alpha^2\beta – \alpha\beta^2 = \dfrac{c\sqrt{b^2 – 4ac}}{a^2}. \]