Question

If $\alpha ,\beta$ are the roots of $a{x}^2+bx+c=0$ then the value $(\frac{\alpha }{\beta }-\frac{\alpha }{\beta }{)}^2$ is:

Answer 1

We have,

$a{x}^2+bx+c=0$

$\alpha and\beta$ be the roots

Then,

$\text{Sums}\text{of}\text{roots}=-\frac{b}{a}$

$\text{Product}\text{of}\text{roots}=\frac{c}{a}$

The value of ${(\frac{\alpha }{\beta }-\frac{\alpha }{\beta })}^2=\left(\frac{\alpha \beta -\beta \alpha }{{\beta }^2}\right)=\frac{0}{{\beta }^2}=0$

Hence, this is the answer.