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.
To find the value of (\(\alpha - \beta\)), we can use the fact that for a quadratic equation of the form ax\(^2\) + bx + c = 0, the sum of the roots is -b/a and the product of the roots is c/a.
In this case, the given quadratic equation is 3x\(^2\) + 4x - 5 = 0, which means that a = 3, b = 4, and c = -5. Using the formula for the sum of the roots, we get:
\(\alpha + \beta = -\frac{b}{a} = -\frac{4}{3}\)
Using the formula for the product of the roots, we get:
\(\alpha \beta = \frac{c}{a} = -\frac{5}{3}\)
To find the value of (\(\alpha - \beta\)), we can use the identity:
\(\alpha - \beta = \sqrt{(\alpha + \beta)^2 - 4\alpha \beta}\)
Plugging in the values we found earlier, we get:
\(\alpha - \beta = \sqrt{\left(-\frac{4}{3}\right)^2 - 4\left(-\frac{5}{3}\right)}\)
Simplifying, we get:
\(\alpha - \beta = \sqrt{\frac{16}{9} + \frac{20}{3}}\)
\(\alpha - \beta = \sqrt{\frac{68}{9}}\)
\(\alpha - \beta = \frac{2\sqrt{17}}{3}\)
Therefore, the value of (\(\alpha - \beta\)) is \(\frac{2\sqrt{17}}{3}\) in surd form.