Question

If α &β are the the zeroes of quadratic polynomial ax²+bx+c so then evaluate α/β+β/α

Answer 1

Given that ,

" α & β are the the zeroes of quadratic polynomial ax²+bx+c so then we need to evaluate α/β+β/α "

Sum of zeroes ,

$\implies \bf \alpha +\beta=\dfrac{-b}{a}...(1)$

Product of zeroes ,

$\implies \bf \alpha.\beta=\dfrac{c}{a}...(2)$

Now our required ,

$\implies \bf \dfrac{\alpha}{\beta}+\dfrac{\beta}{\alpha}\\\\\implies \bf \dfrac{\alpha^2+\beta^2}{\alpha.\beta}\\\\\implies \bf \dfrac{(\alpha^2+\beta^2)-2\alpha.\beta}{\alpha.\beta}\\\\\implies \bf \dfrac{(\dfrac{-b}{a})^2-2.\dfrac{c}{a}}{\dfrac{c}{a}}\;\;\;\;[\;From\ (1) \& (2)\;]\\\\\implies \bf \dfrac{\dfrac{b^2}{a^2}-\dfrac{2c}{a}}{\dfrac{c}{a}}\\\\\implies \bf \dfrac{b^2-2ac}{a^2}*\dfrac{a}{c}\\\\\implies \bf {\blue{\dfrac{b^2-2ac}{c}}}$