Question

if if alpha and beta are the zeros of the quadratic polynomial f x is equal to a x square + bx + c then find the value of x square + beta square if alpha and beta are the zeros of the quadratic polynomial f x is equals to a square + b square + c then find the value of Alpha square plus beta square

Answer 1

i hope this is helpful

Answer 2

Answer:

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

Step-by-step explanation:

Given: α and β are zeroes of f(x)

           f(x) = ax² + bx + c

To find: α² + β²

We use the relation between their coefficient & zeroes, which is given by

$\alpha+\beta=\frac{-b}{a}$

$\alpha\,.\,\beta=\frac{c}{a}$

Now consider the following identity

$(x+y)^2=x^2+y^2+2xy$

put x = α & y = β

we get,

$(\alpha+\beta)^2={\alpha}^2+{\beta}^2+2\,\alpha\,.\beta$

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

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

${\alpha}^2+{\beta}^2=\frac{b^2}{a^2}-2.\frac{c}{a}$

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

Therefore, ${\alpha}^2+{\beta}^2=\frac{b^2-2ca}{a^2}$