Question

if alpha and beta are the zeroes of the polynomial ax^2+bx+c=0 then evaluate alpha square - beta square

Answer 1

The polynomial is :-

ax²+bx+c=0

From the relation between zeroes and coefficients if polynomial we know that :-

Sum of zeroes of a polynomial = (-b)/a

Product of zeroes of a polynomial= c/a

Now,

We have the following :-

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

Putting values:-

$\frac{ - b}{a}( \sqrt{( { \frac{ - b}{a})}^{2} - \frac{4c}{a} } \\ \\ \\ = \frac{ - b}{a} \sqrt{ \frac{ {b}^{2} }{ {a}^{2} } - \frac{4ac}{{a}^{2}}} \\ \\ \\ = \frac{ - b}{ {a}^{2} } \sqrt{ {b}^{2} - 4ac}$

Answer!