If \(\alpha\) and \(\beta\) are the roots of the equation 3x\(^2\) + 4x - 5 = 0, find the value of (\(\alpha - \beta\)), leaving the answer in surd form.
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 (\(\alpha\) - \(\beta\)) = \(\sqrt{\frac{76}{9}}\) = \(\pm \frac{2\sqrt{19}}{3}\)