Question

if alpha and beta are the roots of a quadratic equation ax^2+bx+c=0 tben value of alpha - beta is​

Answer 1

★${\green{\underline{\huge{\mathbb{Solution:-}}}}}$

$\alpha \: and \: \beta \: are \: the \: roots \: of \: a \: quadratic \: equation.$

So,

$\alpha + \beta = \frac{ - b}{a} ........(i)$

$\alpha \beta = \frac{c}{a} ..........(ii)$

• Take (i) no. equation.

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

Squre in both sides.

${( \alpha + \beta )}^{2} = { (\frac{ - b}{a} )}^{2} \\ = > {( \alpha - \beta )}^{2} + 4 \alpha \beta = \frac{ {b}^{2} }{ {a}^{2} } \\ = > {( \alpha - \beta )}^{2} + 4 \times \frac{c}{a} = \frac{ {b}^{2} }{ {a}^{2} } \\ = > {( \alpha - \beta )}^{2} = \frac{ {b}^{2} }{ {a}^{2} } - \frac{4c}{a} \\ = > {( \alpha - \beta )}^{2} = \frac{ {b}^{2} - 4ac }{ {a}^{2} } \\ = ( \alpha - \beta ) = \sqrt{ \frac{ {b}^{2} - 4ac}{ {a}^{2} } } \\ = > ( \alpha - \beta ) = \frac{ \sqrt{ {b}^{2} - 4ac } }{a}$

★★${\green{\underline{\huge{\mathbb{Answer:-}}}}}$

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