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.
Question 1 Report
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.
Answer Details
For \(3x^2+4x-5=0\) with roots \(\alpha,\beta\): sum \(\alpha+\beta=-\dfrac{4}{3}\), product \(\alpha\beta=-\dfrac{5}{3}\).
Use \((\alpha-\beta)^2=(\alpha+\beta)^2-4\alpha\beta\):
\[(\alpha-\beta)^2=\left(-\frac43\right)^2-4\left(-\frac53\right)=\frac{16}{9}+\frac{20}{3}=\frac{16}{9}+\frac{60}{9}=\frac{76}{9}.\] \[\alpha-\beta=\sqrt{\frac{76}{9}}=\frac{\sqrt{76}}{3}=\frac{2\sqrt{19}}{3}.\]So \(\alpha-\beta=\dfrac{2\sqrt{19}}{3}\) (taking the positive surd).
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