Question

If α and β are the zeroes of the polynomial ax2 + bx + c, find the value of α2 + β2​

Answer 1

Given:-

$\rm \alpha \ and \ \beta \ are \ the \ zeroes \ of \ the \ polynomial$

$\rm ax^2+bx+c$

To find:-

$\rm {\alpha}^2 + {\beta}^2$

Solution:-

$\rm\alpha + \beta = \dfrac{-b}{a}$

$\rm\alpha\beta=\dfrac{c}{a}$

We know that,

$\rm{\alpha}^2+\beta ^2 = ( \alpha+\beta)^2 - 2\alpha\beta$

$\rm\therefore\qquad {\alpha}^2+\beta ^2 = \Bigg ( \dfrac{-b}{a}\Bigg )^2 - 2\Bigg ( \dfrac{c}{a}\Bigg )$

$\rm\implies {\alpha}^2+\beta ^2 =\dfrac{b^2}{a^2}- \dfrac{2c}{a}$

$\rm\implies {\alpha}^2+\beta ^2 =\dfrac{b^2-2ac}{a^2}$

Hope it helps...