To solve the quadratic equation x² + 4x + 5 by graphing, we need to first rewrite the equation in a form we can analyze. The equation we will graph is:
y = x² + 4x + 5
Next, it’s helpful to determine the key features of the quadratic function:
- Vertex: The vertex can be found using the formula
x = -b/2a. Here,a = 1andb = 4. - Axis of Symmetry: The axis of symmetry can be found at the line
x = -2. - Value of y at the vertex: Plugging
x = -2back into the equation, we find y = (-2)² + 4(-2) + 5 = 4 – 8 + 5 = 1. Thus, the vertex is at the point (-2, 1).
Now, identifying where the graph intersects the x-axis will help us find the solutions to the equation. However, first, we need to check if there are real x-intercepts:
- For the quadratic to have x-intercepts, we can use the discriminant
D = b² - 4ac. Here,D = 4² - 4(1)(5) = 16 - 20 = -4.
Since the discriminant is negative, this tells us that there are no real solutions, as the graph does not intersect the x-axis.
To graph the function, plot the vertex at (-2, 1). Choose a few more values for x, plug them into the equation to find the corresponding y-values, and plot these points. You will see that the graph opens upwards (since the coefficient of x² is positive) and does not cross the x-axis.
In conclusion, while we can identify the vertex and plot several points, the graph of y = x² + 4x + 5 confirms that there are no real solutions to the equation x² + 4x + 5 = 0.