Question

conclusion is supported by the graph?

• the equation \\(0.5x^2 + 3 = -4x^2 + 24x - 35\\) has no real solutions
• the equation \\(0.5x^2 + 3 = -4x^2 + 24x - 35\\) has one real solution
• the equation \\(0.5x^2 + 3 = -4x^2 + 24x - 35\\) has two real solutions
• the equation \\(0.5x^2 + 3 = -4x^2 + 24x - 35\\) has three real solutions

Explanation:

Identify the functions represented by the graphs

The graph shows two parabolas:

1. An upward-opening parabola with vertex at \((0, 3)\), which corresponds to the function:

$$y_1 = 0.5x^2 + 3$$

1. A downward-opening parabola with vertex at \((3, 1)\), which corresponds to the function:

$$y_2 = -4(x-3)^2 + 1 = -4(x^2 - 6x + 9) + 1 = -4x^2 + 24x - 35$$

Analyze the intersection of the two graphs

The equation \(0.5x^2 + 3 = -4x^2 + 24x - 35\) represents the condition where the two functions intersect:

$$y_1 = y_2$$

Looking at the graph, the two parabolas do not intersect or touch at any point.

Determine the number of real solutions

Since the graphs of the two functions have no points of intersection, the system of equations has no real solutions. Therefore, the equation:

$$0.5x^2 + 3 = -4x^2 + 24x - 35$$

has no real solutions.

Answer:

• The equation \(0.5x^2 + 3 = -4x^2 + 24x - 35\) has no real solutions (Correct answer)
• The equation \(0.5x^2 + 3 = -4x^2 + 24x - 35\) has one real solution
• The equation \(0.5x^2 + 3 = -4x^2 + 24x - 35\) has two real solutions
• The equation \(0.5x^2 + 3 = -4x^2 + 24x - 35\) has three real solutions