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 $m-n$?
To find the $x$-coordinates of the points of intersection between the graphs of $y=x^4$ and $y=5x^2-6+4x^2$, we need to set the two equations equal to each other and solve for $x$. Setting $x^4 = 5x^2-6+4x^2$, we can combine like terms to obtain $x^4 - 9x^2 + 6 = 0$. This is a quadratic equation in terms of $x^2$, so we can use the quadratic formula to solve for $x^2$:x2=−(−9)±√(−9)2−4(1)(6)2(1)Simplifying further, we have:x2=9±√81−242x2=9±√572Since we are only interested in real solutions, we can discard the negative square root. Thus, we have:x2=9+√572To find the values of $m$ and $n$, we need to determine which value is greater. Let's denote $\sqrt{m}$ as the larger solution and $\sqrt{n}$ as the smaller solution. Therefore, we have:√m=√9+√572√n=√9−√572To simplify these expressions, we can rationalize the denominators by multiplying both the numerator and denominator by the conjugate of the denominator. This gives us:√m=√9+√572⋅√2√2√n=√9−√572⋅√2√2Simplifying further, we have:√m=√(9+√57)⋅2√2√n=√(9−√57)⋅2√2Now, let's simplify the expressions under the square roots:√(9+√57)⋅2=√18+2√57√(9−√57)⋅2=√18−2√57Therefore, we have:√m=√18+2√57√2√n=√18−2√57√2To determine the value of $m-n$, we need to subtract $\sqrt{n}$ from $\sqrt{m}$:m−n=(√18+2√57√2)2−(√18−2√57√2)2Expanding and simplifying, we have:m−n=(18+2√57)2−(18−2√57)2m−n=2√572m−n=√57Therefore, the value of $m-n$ is $\sqrt{57}$.
