Question #40442

This question constructs a 'Lethal Collision' between a bounded trigonometric function and a quadratic polynomial.

The core trap lies in the boundary conditions of the modulus inequality.

Students frequently calculate the maximum of the LHS ($2\sqrt{2}$) and the minimum of the RHS ($2\sqrt{2}$ for $a > 0$), concluding that solutions exist for $a \ge 0$ by simply equating the bounds.

However, they fail to verify if the domain values ($x$-coordinates) where these bounds are achieved actually coincide.

Since the RHS minimum occurs at $x=1$ and the LHS maximum occurs at $x = 5\pi/12 + n\pi/2$, they never simultaneously reach $2\sqrt{2}$.

The true solution requires applying the Intermediate Value Theorem for $a < 0$, where the inverted parabola must intersect the bounded trigonometric wave.