1. **State the problem:** We are given the quadratic function $$c(x) = -x^2 + 11x - 24$$ and three statements about it: - #1: The x-intercepts are (3,0) and (7,0). - #2: The maximum value is 6.25. - #3: The vertex is in quadrant I. 2. **Find the x-intercepts:** To find x-intercepts, solve $$c(x) = 0$$: $$-x^2 + 11x - 24 = 0$$ Multiply both sides by $$-1$$ to simplify: $$\cancel{-1}(-x^2 + 11x - 24) = \cancel{-1}(0)$$ $$x^2 - 11x + 24 = 0$$ Factor the quadratic: $$(x - 3)(x - 8) = 0$$ So, $$x = 3$$ or $$x = 8$$. Therefore, the x-intercepts are (3,0) and (8,0), not (7,0). 3. **Find the vertex:** The vertex x-coordinate is the midpoint of the roots: $$x_v = \frac{3 + 8}{2} = \frac{11}{2} = 5.5$$ Find the y-coordinate by evaluating $$c(5.5)$$: $$c(5.5) = -(5.5)^2 + 11(5.5) - 24 = -30.25 + 60.5 - 24 = 6.25$$ 4. **Determine the quadrant of the vertex:** The vertex is at (5.5, 6.25). Since $$x > 0$$ and $$y > 0$$, the vertex lies in quadrant I. 5. **Check the maximum value:** Since the parabola opens downward (coefficient of $$x^2$$ is negative), the vertex y-value is the maximum value, which is 6.25. **Summary:** - Statement #1 is incorrect because the second x-intercept is (8,0), not (7,0). - Statement #2 is correct. - Statement #3 is correct. **Final answer:** Only statement #1 is incorrect.