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.
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Correct Answer: Option
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
(\(\alpha\) - \(\beta\)) = \(\sqrt{\frac{76}{9}}\)
= \(\pm \frac{2\sqrt{19}}{3}\)
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
(\(\alpha\) - \(\beta\)) = \(\sqrt{\frac{76}{9}}\)
= \(\pm \frac{2\sqrt{19}}{3}\)