Question 6
$\alpha$ and $\beta$ satisfy $\dfrac{\alpha^2-\alpha\beta+\beta^2}{\alpha^2+\alpha\beta+\beta^2}=\dfrac{31}{43}$ and $\dfrac1\alpha+\dfrac1\beta=\dfrac72$. The quadratic equation with roots $\alpha,\beta$ is $px^2+qx+r=0$, where $p,q,r$ are integers, $p\gt 0$, and $\gcd(p,q,r)=1$. Find $p+q+r$.
The source displays a quadratic expression; it is understood to be set equal to zero. Its printed key conflicts with the displayed algebraic conditions, as explained in the solution.
The source displays a quadratic expression; it is understood to be set equal to zero. Its printed key conflicts with the displayed algebraic conditions, as explained in the solution.