If the sum of the roots of the quadratic equation ax^2+bx+c=0 is equal to the sum of the s…

Question

If the sum of the roots of the quadratic equation $ax^2+bx+c=0$ is equal to the sum of the squares of their reciprocals, then $\dfrac ac,\dfrac ba,\dfrac cb$ are in:

Accepted Answer

$\alpha+\beta=-\dfrac ba,\ \alpha\beta=\dfrac ca$. Given $\alpha+\beta=\dfrac{1}{\alpha^2}+\dfrac{1}{\beta^2}=\dfrac{\alpha^2+\beta^2}{\alpha^2\beta^2}$:

$$-\frac ba=\frac{b^2/a^2-2c/a}{c^2/a^2}.$$

Simplifying gives $2a^2c=ab^2+bc^2$, i.e. (dividing by $abc$) $\dfrac{2a}{b}=\dfrac ca+\dfrac bc$, so $\dfrac ca,\dfrac ab,\dfrac bc$ are in A.P., which means $\dfrac ac,\dfrac ba,\dfrac cb$ are in H.P.

Solutions · 1