If \(\alpha\) and \(\beta\) are the roots of the equation 3x\(^2\) + 4x - 5

Explanation

Note that \(\alpha\) + \(\beta\) = - \(\frac{4}{3}\) and (\(\alpha - \beta\))\(^2\)

= a\(^2\) + \(\beta\)\(^2\) - 2\(\alpha\beta\) = (\(\alpha + \beta\))\(^2\) - 4\(\alpha \beta\)

Thus, substituting for \(\alpha + \beta\) and \(\alpha \beta\)

simplify to get (\(\alpha - \beta\))\(^2\) = (-\(\frac{4}{3}\))\(^2\) - 4(-\(\frac{5}{3}\))

= \(\frac{16}{9} + \frac{20}{3}\)

= \(\frac{16+60}{9}\)

= \(\frac{76}{9}\)

Taking square root of both sides

(\(\alpha\) - \(\beta\)) = \(\sqrt{\frac{76}{9}}\)

= \(\pm \frac{2\sqrt{19}}{3}\)