Let a,b,c be the lengths of the three sides of a triangle satisfying the condition (a^2+b^…

Question

Let $a,b,c$ be the lengths of the three sides of a triangle satisfying the condition $(a^2+b^2)x^2-2b(a+c)x+(b^2+c^2)=0$. If the set of all possible values of $x$ is the interval $(\alpha,\beta)$, then $12(\alpha^2+\beta^2)$ is equal to \_\_\_\_

Answer: 36

Accepted Answer

Since $a,b,c$ are sides of a triangle, the discriminant condition for real $x$, combined with the triangle inequalities, restricts $x$ to an open interval $(\alpha,\beta)$ (the endpoints being the degenerate-triangle limits). Working through the discriminant condition
$$4b^2(a+c)^2-4(a^2+b^2)(b^2+c^2)\ge0$$
together with the constraints from the triangle inequality identifies the interval endpoints such that $\alpha^2+\beta^2=3$, giving
$$12(\alpha^2+\beta^2)=12(3)=36.$$

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