To solve the equation x2 + 4x + 5 = 0 by graphing, you’ll first need to understand the process of converting the quadratic equation into a function that can be plotted. Here’s a step-by-step guide:
- Define the function: Rewrite the equation in the standard quadratic function format. In this case, you can express it as f(x) = x2 + 4x + 5.
- Determine the vertex: The vertex of a parabola described by the function ax2 + bx + c can be found using the formula x = -b/(2a). For our function, a = 1 and b = 4. Thus, the x-coordinate of the vertex is:
- x = -4/(2 * 1) = -2
- Next, plug this x-value back into the function to find the y-coordinate:
- f(-2) = (-2)2 + 4(-2) + 5 = 4 – 8 + 5 = 1
- So, the vertex of the parabola is at the point (-2, 1).
- Identify the direction of the parabola: Since the coefficient of x2 (which is 1) is positive, the parabola opens upwards.
- Find the y-intercept: To find where the parabola intersects the y-axis, set x = 0.
- f(0) = 02 + 4(0) + 5 = 5
- So, the y-intercept is at the point (0, 5).
- Plot the points: Use the vertex (-2, 1) and y-intercept (0, 5) to begin sketching the graph. You can also calculate additional points for a smoother curve by choosing x-values around the vertex, such as -1 and -3:
- f(-1) = (-1)2 + 4(-1) + 5 = 2
- f(-3) = (-3)2 + 4(-3) + 5 = 2
- Now you have points (-1, 2) and (-3, 2) to plot as well.
- Sketch the graph: Draw a smooth curve through the points plotted. The graph should be a U-shaped curve, with the lowest point at the vertex (-2, 1).
- Determine the x-intercepts: Since the problem involves finding solutions to the equation, check where the graph intersects the x-axis. In this case, you can see that the parabola does not cross the x-axis (the vertex is above zero, and there are no real roots) indicating that the equation has no real solutions.
In conclusion, by graphing the function f(x) = x2 + 4x + 5, you discover that it does not intersect the x-axis, meaning there are no real solutions to the equation x2 + 4x + 5 = 0.