Question bank: Second Degree Equation: Coefficients

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.

A ball is thrown vertically upwards from the ground, reaching a maximum height of 16 meters above the ground after 1 second of being thrown. The height h(t) of the ball at time t, in seconds, is given by the equation h(t) = -5t^2 + 10t, where h(t) is the height in meters. Considering the equation of the ball's motion, identify and explain the meaning of the coefficients a, b, and c in the general equation of a parabola y = ax^2 + bx + c. Then, calculate the value of a + b + c and determine if the sum of the roots of the equation h(t) = 0 is equal, greater, or less than 1, justifying your answer based on the coefficients of the equation h(t).

In an amusement park, the trajectory of a free fall ride is represented by a parabola. Its highest point is 100 meters and the ride reaches the ground after 5 seconds. Considering the acceleration due to gravity as -10 m/s², and that the movement of the ride starts from rest and from the ground, determine the equation of the ride's motion, expressed in the general form of the second-degree equation.

João is a young entrepreneur who opened his own video game store. The number of video games sold per month can be represented by a quadratic equation, where the coefficient 'a' varies depending on the time of year. In the store's anniversary month, the coefficient 'a' is 3, and he decides to give a 10% discount to customers. It is known that the relationship between the number of sales (v) and the price (p) of video games with the discount is given by v = ap².