Question

if a(alpha) and b(beta) are the roots of quadratic equation=
ax^2+bx+c=0. then find the value of a(alpha)^2-b(beta)^2​

Answer 1

➠Given equation=

$a {x}^{2} + bx + c = 0\\ ➠\alpha + \beta = \frac{ - b}{a} \\➠ \alpha \beta = \frac{c}{a}$

${α}^{2} - {β }^{2} \\➠ \: (α + β)( \alpha - \beta ) \\ ( \alpha + \beta ) \sqrt{ {( \alpha + \beta )}^{2} - 4 \alpha \beta } \\ ➠ .....using \: identity \\➠ \frac{ - b}{a} \sqrt{ { (\frac{ - b}{a} })^{2} - 4 \frac{c}{a} } \\ \frac{ - b}{a} \sqrt{ \frac{ {b}^{2} - 4ac }{ {a}^{2} } } \\ ➠ \frac{ - b}{ {a}^{2} } \sqrt{ {b}^{2} - 4ac}$

➠Original answer☑

Answer 2

The above answer is correct