Which of the following accurately describes what the graph of y=5x2+50x+125 looks like, with the repeated root of x=−5 ?

Question

Which of the following accurately describes what the graph of  y=5x2+50x+125  looks like, with the repeated root of  x=−5 ?

Statement #1: The graph opens downward.

Statement #2: The graph has a vertex to the right of the x-axis.

Statement #3: The graph touches the x-axis once.

GPT-4o mini · Answer

Let's analyze the equation $ y = 5x^2 + 50x + 125 $ and its properties, particularly focusing on the repeated root $ x = -5 $. 1. **Determining the Direction of the Graph:** The coefficient of $ x^2 $ is $ 5 $, which is positive. Therefore, the graph opens **upward**. This means Statement #1 is **false**. 2. **Finding the Vertex of the Parabola:** To find the vertex, we can use the formula for the x-coordinate of the vertex in a quadratic equation $ ax^2 + bx + c $, given by $ x = -\frac{b}{2a} $. Here, $ a = 5 $ and $ b = 50 $: $$ x = -\frac{50}{2 \times 5} = -\frac{50}{10} = -5 $$ The vertex is at $ x = -5 $. This is exactly at the repeated root, and thus is not to the right of the x-axis. Therefore, Statement #2 is **false**. 3. **Examining the Root:** Since $ x = -5 $ is a repeated root, the graph touches the x-axis at this point and does not cross it. Therefore, the graph touches the x-axis only once, which makes Statement #3 **true**. In summary: - Statement #1: **False** (the graph opens upward) - Statement #2: **False** (the vertex is at $ x = -5 $) - Statement #3: **True** (the graph touches the x-axis at one point, $ x = -5 $)