If $\alpha$ and $\beta$ are the zeroes of the polynomial $ax^2+bx+c$, find the value of $\alpha^2+\beta^2$.

Given: $\alpha$ and $\beta$ are the zeroes of the polynomial $ax^2+bx+c$.

To do: To find the value of $\alpha^2+\beta^2$.

Solution:

As given $\alpha$ and $\beta$ are the zeroes of the polynomial $ax^2+bx+c$.

Sum of the zeroes $\alpha+\beta=-\frac{b}{a}$

Product of the zeroes $\alpha\beta=\frac{c}{a}$

$\because ( \alpha+\beta)^2=\alpha^2+\beta^2+2\alpha\beta$

$\Rightarrow \alpha^2+\beta^2=( \alpha+\beta)^2-2\alpha\beta$

$\Rightarrow \alpha^2+\beta^2=( -\frac{b}{a})^2-2\times\frac{c}{a}$

$\Rightarrow \alpha^2+\beta^2=\frac{b^2}{a^2})-2\times\frac{c}{a}$

$\Rightarrow \alpha^2+\beta^2=\frac{ab^2-2a^2c}{a^2}$

$\Rightarrow \alpha^2+\beta^2=\frac{b^2-2ac}{a}$