Answer to Question #189473 in Math for Charles

Question

If α+β are the roots of the equation 3x^2-5x+6=0. Find the value of α/β+β/α

Answer 1

Let $\alpha$ and $\beta$ be roots of the equation $3x^2-5x+6=0$ ,

Using Viet Theorem, we obtain that $\alpha +\beta =\frac{5}{3}$ and $\alpha \beta=\frac{6}{3}=2$ .

$\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^2+\beta^2}{\alpha\beta}= \frac{\alpha^2+2\alpha\beta+\beta^2-2\alpha \beta}{\alpha\beta}= \frac{(\alpha+\beta)^2-2\alpha \beta}{\alpha\beta}=\frac{\left(\frac{5}{3}\right)^2-2\cdot 2}{2}=\frac{25-36}{18}=-\frac{11}{18}$

Answer: $\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=-\frac{11}{18}$ .