Question

Let $a$, $b$, $c$ be the lengths of 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 1

The given quadratic equation in $x$ is:

$(a^2 + b^2)x^2 - 2b(a + c)x + (b^2 + c^2) = 0.$

This can be written in the form:

$(ax - b)^2 + (bx - c)^2 = 0.$

Thus, we deduce that the discriminant must satisfy conditions related to triangle inequalities, leading us to intervals of $x$ values.

By evaluating the possible values of $x$, we find that the interval $(\alpha, \beta)$ corresponds to:

$\alpha = \frac{1 - \sqrt{5}}{2}, \quad \beta = \frac{1 + \sqrt{5}}{2}.$

Then, calculate $12(\alpha^2 + \beta^2)$:

$12(\alpha^2 + \beta^2) = 36.$

Thus, the answer is:

36\.