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(alphabeta) = ((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{2sqrt{19}}{3})