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

Correct Answer: 36

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:

\[ 36. \]