To find the solutions to the equation x² + 4x + 4 – 2x – 1 = 0, we first simplify the equation. This can be rewritten as:
- x² + (4 – 2)x + (4 – 1) = 0
- x² + 2x + 3 = 0
This is a quadratic equation of the form ax² + bx + c = 0, where a = 1, b = 2, and c = 3.
The graph that can be used to visualize the solutions to this quadratic equation is a parabola. Specifically, the graph of the function f(x) = x² + 2x + 3 will show the shape of the parabola, which opens upwards.
To find the solutions (or roots) of the equation, we can look for the points where the parabola intersects the x-axis. However, to determine whether it intersects the x-axis, we can use the discriminant of the quadratic equation:
Discriminant (D) = b² – 4ac
Substituting in our values, we have:
- D = (2)² – 4(1)(3) = 4 – 12 = -8
Since the discriminant is negative (D < 0), this means that the parabola does not intersect the x-axis and thus does not have real solutions. Instead, the solutions will be complex numbers.
In conclusion, while the parabola is the graph used to investigate the solutions of the quadratic equation, it will not touch the x-axis, indicating that there are no real solutions to the equation x² + 2x + 3 = 0.