To analyze the function \( y = 5x^2 + 50x + 125 \), let's consider the following aspects:
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Nature of the graph: The function is a quadratic function of the form \( y = ax^2 + bx + c \) where \( a = 5 \), \( b = 50 \), and \( c = 125 \). Since \( a > 0 \), the graph opens upward.
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Roots of the graph: You mentioned that the function has a repeated root at \( x = -5 \). If there is a repeated root, it means that the discriminant of the quadratic \( b^2 - 4ac \) is zero. We can check if \( x = -5 \) is indeed a root.
- Calculate \( b^2 - 4ac \):
- \( b^2 = 50^2 = 2500 \)
- \( 4ac = 4 \cdot 5 \cdot 125 = 2500 \)
- Therefore, \( b^2 - 4ac = 2500 - 2500 = 0 \). This confirms that \( x = -5 \) is a repeated root.
- Calculate \( b^2 - 4ac \):
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Behavior of the graph around the root: Since \( x = -5 \) is a repeated root, the graph touches the x-axis at this point and does not cross it.
Based on these insights, let's now evaluate the statements:
- A: The graph opens downward: False (it opens upward).
- B: The graph has a vertex to the right of the x-axis: This is unclear without calculation, but since it touches the x-axis at \( x = -5 \), the vertex will be located at \( x = -5 \) (which is not to the right of the x-axis).
- C: The graph touches the x-axis once: True (it touches at the repeated root \( x = -5 \)).
Thus, the correct description is:
C: The graph touches the x-axis once.