Final answer:
The area of the triangle formed by the x- and y-intercepts of the parabola y = 0.5(x - 3)(x + k) is given as 1.5 square units. By solving the equation for the area of the triangle, we find two possible values for k, which are k = -2 and k = 1.
Step-by-step explanation:
The area of the triangle formed by the x- and y-intercepts of the parabola y = 0.5(x − 3)(x + k) is equal to 1.5 square units. The x-intercepts are the solutions to the equation when y is set to 0, which gives us x = 3 and x = -k as the intercepts. The y-intercept is found by setting x = 0, which gives us y = −1.5k. The area of a triangle is given by the formula 0.5 * base * height. Here, the base is 3 - (-k) = 3 + k, and the height is y-intercept = −1.5k.
Thus, the area A becomes 0.5 * (3 + k) * −1.5k = 1.5. We simplify this equation to find the value(s) of k that solve the problem.
Solving the Equation:
0.5 * (3 + k) * −1.5k = 1.5
−0.75k * (3 + k) = 1.5
−0.75k2 - 2.25k + 1.5 = 0
If we divide through by −0.75, we obtain a quadratic equation which can be solved using the quadratic formula or by factoring:
k2 + 3k - 2 = 0
(k + 2)(k - 1) = 0
Thus, the values of k are k = -2 and k = 1. Therefore, there are two possible values for k that will give the area of the triangle as 1.5 square units.