(Originais Teachy 2023) - Question Hard of Mathematics

Consider a second degree equation in general form ax^2 + bx + c = 0, where a, b, and c are real coefficients and a ≠ 0. Let r1 and r2 be the real roots of this equation. The sum of the roots, S, is given by S = -b/a and the product of the roots, P, is given by P = c/a. In a context where the equation models the trajectory of a projectile in a specific gravitational field, it is known that the sum of the roots represents the point of maximum height reached by the projectile and the product of the roots is related to the horizontal distance traveled by the projectile during its flight. Given a constant value k > 0 and knowing that the equation a(x - r1)(x - r2) = k, where r1 and r2 are the roots of the equation, answer: (1) What is the relationship between the value of k and the maximum height reached by the projectile? (2) Considering that the projectile is launched from an initial height h0 and that the equation of the parabolic motion is given by -g/2 * t^2 + v0t + h0, where g is the acceleration due to gravity and v0 is the initial velocity of the projectile, how are the sum and the product of the roots of the second degree equation related to the maximum height and the horizontal range of the projectile, respectively? Explain the mathematical and physical reasoning involved, highlighting the importance of the coefficients and roots in the context of the problem.