Evaluation Using Zeros of a Quadratic Polynomial
Video Explanation
Question
If \( \alpha \) and \( \beta \) are the zeroes of the polynomial
\[ f(x) = ax^2 + bx + c, \]
find
\[ \frac{1}{\alpha^2} + \frac{1}{\beta^2}. \]
Solution
Step 1: Write Relations Between Zeroes 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
\[ \frac{1}{\alpha^2} + \frac{1}{\beta^2} = \left(\frac{1}{\alpha} + \frac{1}{\beta}\right)^2 – \frac{2}{\alpha\beta} \]
Step 3: Evaluate Each Term
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha\beta} = \frac{-\frac{b}{a}}{\frac{c}{a}} = -\frac{b}{c} \]
\[ \frac{2}{\alpha\beta} = \frac{2a}{c} \]
Step 4: Substitute and Simplify
\[ \frac{1}{\alpha^2} + \frac{1}{\beta^2} = \left(-\frac{b}{c}\right)^2 – \frac{2a}{c} \]
\[ = \frac{b^2}{c^2} – \frac{2a}{c} \]
Conclusion
\[ \boxed{ \frac{1}{\alpha^2} + \frac{1}{\beta^2} = \frac{b^2}{c^2} – \frac{2a}{c} } \]