Question

What is α (Alpha) - β (Beta)?

(like, in a quadratic equation: ax² + bx + c, roots "α + β = $\frac{-b}{a}$", and "αβ = $\frac{c}{a}$"; then what is denoted by "α - β = ?"

Answer 1

It is given that

$:  \implies \tt \: \alpha + \beta = - \dfrac{b}{a}$

$:  \implies \tt \: \alpha \beta = \dfrac{c}{a}$

We know that

$:  \implies \tt \: {( \alpha + \beta )}^{2} - {( \alpha - \beta )}^{2} = 4 \alpha \beta$

$:  \implies \tt \: {( \alpha - \beta )}^{2} = {( \alpha + \beta )}^{2} - 4 \alpha \beta$

$:  \implies \tt \: {( \alpha - \beta )}^{2} = \dfrac{ {b}^{2} }{ {a}^{2} } - \dfrac{4c}{a}$

$:  \implies \tt \: {( \alpha - \beta )}^{2} = \dfrac{ {b}^{2} - 4ac }{ {a}^{2} }$

$:  \implies \tt \: \alpha - \beta = \pm \: \dfrac{ \sqrt{ {b}^{2} - 4ac} }{a}$