Question

find the value of alpha and beta from poly nomial ax²+bx+c then find alpha ²+beta²

Answer 1

The polynomial is
P(x) = ax²+bx+c

Alpha and beta are the zeroes of the polynomial.
For quadratic polynomials the relation between the zeroes and the coefficients is
$\alpha \beta = \frac{c}{a} \\ \alpha + \beta = \frac{ - b}{a}$
We need to find value of alpha²+beta²

${( \alpha + \beta )}^{2} = { \alpha }^{2} + { \beta }^{2} + 2 \alpha \beta \\ { \alpha }^{2} + { \beta }^{2} = {( \alpha + \beta) }^{2} - 2 \alpha \beta \\ \\ putting \: the \: values \\ { \alpha }^{2} + { \beta }^{2} = {( \frac{ - b}{a}) }^{2} - \frac{2c}{a} = \frac{ {b}^{2} }{ {a}^{2} } - \frac{2c}{a} \\ = \frac{ {b}^{2} - 2ac}{ {a}^{2} }$
Hope this will be helping you....