The graphs of $y=x^4$ and $y=7x^2-10$ intersect at four points with $x$-coordinates $\pm \sqrt{m}$ and $\pm \sqrt{n}$, where $m > n$. What is $m-n$?
I need to find the values of $m$ and $n$ that satisfy both equations simultaneously. I can start by equating $y=x^4$ and $y=7x^2-10$ and simplifying. This gives me $x^4-7x^2+10=0$. I notice that this is a quartic equation, but it has only even powers of $x$. This means I can substitute $u=x^2$ and get a quadratic equation in $u$. This gives me $u^2-7u+10=0$. I can factor this equation as $(u-2)(u-5)=0$. So $u=2$ or $u=5$. But $u=x^2$, so I need to take the square root of both sides. This gives me $x=\pm \sqrt{2}$ or $x=\pm \sqrt{5}$. These are the four $x$-coordinates of the intersection points. Therefore, $m=5$ and $n=2$. And $m-n=5-2=3$. 