To identify the correct information about the quadratic expression \(7x^2 - 4x + 12\), we can analyze various aspects such as its standard form, the coefficients, and its properties.
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Standard Form: The expression is already in standard form, which is \(ax^2 + bx + c\).
- Here, \(a = 7\), \(b = -4\), and \(c = 12\).
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Type: This is a quadratic expression because it is of the form \(ax^2 + bx + c\).
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Vertex and Axis of Symmetry:
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The axis of symmetry can be found using the formula \(x = -\frac{b}{2a}\).
- \(x = -\frac{-4}{2 \cdot 7} = \frac{4}{14} = \frac{2}{7}\).
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The vertex can be found by plugging this x-value back into the expression:
- \(y = 7\left(\frac{2}{7}\right)^2 - 4\left(\frac{2}{7}\right) + 12\).
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Discriminant: The discriminant \(D\) of the quadratic can be calculated using the formula \(D = b^2 - 4ac\).
- \(D = (-4)^2 - 4(7)(12) = 16 - 336 = -320\).
- Since the discriminant is negative, this means the quadratic has no real roots (the parabola does not intersect the x-axis).
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Graph: The graph of this quadratic will be a parabola that opens upwards (since \(a > 0\)) and will be positioned above the x-axis since it has no real roots.
Given these points, here’s a correct list summarizing the key characteristics of the expression \(7x^2 - 4x + 12\):
- Type: Quadratic expression
- Standard Form: \(7x^2 - 4x + 12\)
- Coefficients: \(a = 7\), \(b = -4\), \(c = 12\)
- Discriminant: \(D = -320\) (no real roots)
- Axis of Symmetry: \(x = \frac{2}{7}\)
- Vertex: Located at \(\left(\frac{2}{7}, y\right)\) where \(y\) can be calculated.
- Direction: Opens upwards.
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