Question

Let $\alpha ,\beta$ be two roots of the quadratic equation $x^2+4x+1=0$ then find the value of ${\alpha }^2+{\beta }^2$.

Answer 1

Given $\alpha ,\beta$ be two roots of the quadratic equation $x^2+4x+1=0$.

Then from the relation between the roots and the co-efficient we get,

$\alpha +\beta =-4$.....(1) and $\alpha .\beta =1$.....(2).

Now,

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

${\alpha }^2+{\beta }^2$$=(-4{)}^2-2.1$

${\alpha }^2+{\beta }^2$$=16-2$ [ using (1) and (2)]

${\alpha }^2+{\beta }^2$$=14$.