<s>[INST] How many times do these two graphs intersect for values x >0? When the curves
$$y = x^2 + 4x -5$$
and
$$y = \frac{1}{1+​x^2} $$
are drawn in the $xy$-plane, how many times do the two graphs intersect for values of $x > 0$ ?
I equate the value of $y$, then the equation comes in the fourth power of $x$ . How I can solve this?
Thanks in advance.
 [/INST] While polynomials of degree 4 can be solved by radicals, that is not needed here.
The first graph, $y=x^2+4x-5 = (x+5)(x-1)$ is positive if $x\lt -5$ or if $x\gt 1$. It is increasing on $x\gt 1$. The graph of $y=\frac{1}{1+x^2}$ is always positive, and is decreasing on $x\gt 0$.
When  we look at the portions of the graphs that are on the first quadrant, the graph of $y=x^2+4x-5$ is going up, the graph of $y=\frac{1}{1+x^2}$ is going down.  And $y=x^2+4x-5$ is smaller than $y=\frac{1}{1+x^2}$ when $x=1$, but is larger when $x=2$.
So...
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