To analyze the quadratic function \(y = 5x^2 + 50x + 125\) and the correctness of the statements provided:
1. Check the opening direction of the graph
The quadratic formula is in the form \(y = ax^2 + bx + c\). Here, \(a = 5\) which is positive.
- Conclusion: The graph opens upward (not downward).
2. Determine the vertex location
The vertex of a quadratic function in standard form can be calculated using the formula for the x-coordinate of the vertex: \[ x = -\frac{b}{2a} \] Substituting the values \(a = 5\) and \(b = 50\): \[ x = -\frac{50}{2(5)} = -\frac{50}{10} = -5 \]
The x-coordinate of the vertex is \(-5\), which is to the left of the y-axis and also to the left of the x-axis.
- Conclusion: The graph has a vertex at \(x = -5\), which is not to the right of the x-axis.
3. Analyze the roots
The quadratic equation \(y = 5x^2 + 50x + 125\) can be factored or analyzed to find the roots. Since you mentioned a repeated root at \(x = -5\), we can confirm that by solving:
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Use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Substituting the values: \[ x = \frac{-50 \pm \sqrt{50^2 - 4 \cdot 5 \cdot 125}}{2 \cdot 5} \] Calculating inside the square root: \[ 50^2 = 2500 \] \[ 4 \cdot 5 \cdot 125 = 2500 \] \[ b^2 - 4ac = 2500 - 2500 = 0 \] Since the discriminant is \(0\), there is indeed one repeated root at \(x = -5\).
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Conclusion: The graph touches the x-axis once at the point \((-5, 0)\).
Summary of the Statements
- Statement #1: The graph opens downward. (False)
- Statement #2: The graph has a vertex to the right of the x-axis. (False)
- Statement #3: The graph touches the x-axis once. (True)
Therefore, the only accurate statement is Statement #3.
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