The quadratic equation 2x2 + 4x + 12 = 0 has several important characteristics that can be analyzed using the properties of quadratic equations.
1. **Standard Form**: The equation is in the standard form of a quadratic equation, which is ax2 + bx + c = 0, where:
- a = 2
- b = 4
- c = 12
2. **Discriminant**: The discriminant (D) is calculated as: D = b2 – 4ac. For this equation:
D = 42 – 4(2)(12) = 16 – 96 = -80
The discriminant is negative (D < 0), which indicates that the equation has no real roots but instead has two complex roots.
3. **Vertex**: The vertex of a quadratic equation given by y = ax2 + bx + c can be found using the formula x = -b / (2a). Substituting in our values:
x = -4 / (2 * 2) = -4 / 4 = -1
To find the corresponding y-coordinate, we substitute x = -1 back into the original equation:
y = 2(-1)2 + 4(-1) + 12 = 2(1) – 4 + 12 = 10
Thus, the vertex is (-1, 10).
4. **Axis of Symmetry**: The axis of symmetry for the quadratic equation can be represented by the line x = -b / (2a), which in this case is:
x = -(-4) / (2(2)) = 1
This line of symmetry indicates that the parabola is symmetric about this line.
5. **Graph Orientation**: Since the value of a (2) is positive, the parabola opens upwards. This means that the vertex represents the minimum point of the curve.
Overall, the quadratic equation 2x2 + 4x + 12 = 0 provides rich information about its nature, revealing that it does not cross the x-axis and instead has complex roots, with its vertex at (-1, 10) and an upward-opening parabola.