Problem

The key is to rewrite this as a quadratic and show its discriminant is always positive. Expand both products and combine like terms to get a standard quadratic form, then compute the discriminant $\Delta = B^2 - 4AC$. The trick is recognizing that the ordering $a < b < c < d$ creates a specific algebraic structure—when you substitute the midpoints or use the constraint that $b$ and $d$ are positioned between $a$ and $c$, the discriminant simplifies to an expression that's manifestly positive, guaranteeing two real distinct roots.

The key is to rewrite this as a quadratic and show its discriminant is always positive. Expand both products and combine like terms to get a standard quadratic form, then compute the discriminant $\Delta = B^2 - 4AC$. The trick is recognizing that the ordering $a < b < c < d$ creates a specific algebraic structure—when you substitute the midpoints or use the constraint that $b$ and $d$ are positioned between $a$ and $c$, the discriminant simplifies to an expression that's manifestly positive, guaranteeing two real distinct roots.