The graphs of $y=x^4$ and $y=5x^2-6+4x^2$ intersect at four points with $x$-coordinates $\pm \sqrt{m}$ and $\pm \sqrt{n}$, where $m n$. What is $m-n$?

Question

The graphs of $y=x^4$ and $y=5x^2-6+4x^2$ intersect at four points with $x$-coordinates $\pm \sqrt{m}$ and $\pm \sqrt{n}$, where $m > n$. What is $m-n$?

GPT 3.5 · Answer

Setting the two equations equal and rearranging the terms gives $x^4 = 5x^2 + 4x^2 - 6$. Simplifying gives $x^4 - 9x^2 + 6 = 0$. This equation can be factored as $(x^2-3)(x^2 - 2)$. The four $x$-coordinates where the graphs of $y = x^4$ and $y= 5x^2 + 4x^2 - 6$ intersect are $\pm \sqrt{3}$ and $\pm \sqrt{2}$, so $m-n = \sqrt{3}^2 - \sqrt{2}^2 = 3-2=\boxed{1}$.