The function whon't intercept the x-axis because the minimum is at x = 4.75
Why the function does not cross the x-axis?
A function can cross the x-axis only if the value y = 0 is allowed, and the y-axis only if the value x = 0 is allowed.
Here we have the function:
f(x) = x² + x + 5
When x = 0 we have:
f(0) = 0² + 0 + 5
f(0) = 5
Now, let's see that there is no value of x such that f(x) = 0.
We can see this because the vertex of the function is at:
x = -1/(2*1) = -1/2
Evaluating there, we will get:
f(-1/2) = (-1/2)² + -1/2 + 5
= 1/4 - 1/2 + 5
= 4.75
That is the minimum value that the function can take, so it will never intercept the x-axis.