Positive and negative roots of e to the power x equals to x.
e^x = x
since e^x is always positive, there are no negative roots.
Since the slope of y=x is 1, and the slope of y=e^x is greater than 1 for positive x, then since e^0>0, the two graphs never intersect.
In short, there are no real solutions.
Confirming Steve's conclusion:
https://www.wolframalpha.com/input/?i=Plot+y+%3D+e%5Ex,+y+%3D+x
To find the positive and negative roots of the equation where e^x = x, we can rewrite the equation as e^x - x = 0 and then solve it.
To begin solving, let's plot the graph of the function y = e^x - x to visualize its behavior:
1. Choose a range of x-values that you suspect might contain the roots, for example, x = -5 to x = 5.
2. Calculate the corresponding y-values by substituting each x-value into the equation y = e^x - x.
For example, when x = -5, y = e^(-5) - (-5) = e^(-5) + 5.
3. Plot the points (x, y) on a graph.
Alternatively, you can use a graphing calculator, software like Microsoft Excel, or online tools to plot the graph.
Once you have plotted the graph, you can observe where the curve intersects the x-axis. Those points of intersection represent the roots of the equation e^x = x.
In this case, the positive root is approximately x = 0.56714, and the negative root is approximately x = -1.67835.