Evaluation of \( \alpha – \beta \)

Video Explanation

Question

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

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

evaluate \( \alpha – \beta \).

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: Use the Identity for Difference of Zeros

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

Substitute the values:

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

\[ = \frac{b^2 – 4ac}{a^2} \]

Step 3: Take Square Root

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

Conclusion

The required value is:

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

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