1) Given that \(x^2 + \lfloor x \rfloor = 75,\)find all possible values for $x.$
2) Let \((x_1,y_1),(x_2,y_2),\dots,(x_n,y_n)\) be the real solutions to \(\begin{align*} |y| - \frac{2x}{|x|} &= -1, \\ x|x| + y|y| &= 24. \end{align*}\)Find \(x_1 + y_1 + x_2 + y_2 + \dots + x_n + y_n.\)
3) Let \(f(x) = \begin{cases} x^2+2 &\text{if } x If the graph y=f(x) is continuous, find the sum of all possible values of n.
+6248 1)
\(\text{well we can see without much difficulty that $x \in (8, 9)$ somewhere}\\ \text{in that range $\lfloor x \rfloor = 8$}\\ \text{So it must be that}\\ x^2 = 75-8 = 67\\ x = \sqrt{67}\)
2)
\(|y| - \dfrac{2x}{|x|} = -1\\ |y|-2\text{sgn$(x)= -1$}\\ |y|=-1 + 2\text{sgn$(x)$}\\ \text{The only solutions to this are }\\ x>0,~y = \pm 1\)
\(\text{applying the results of the previous equation to the second}\\ x^2 +1 = 24\\ x^2 -1 = 24\\ x =\sqrt{23}, ~ 5\\ \text{the ordered pairs of solutions are }\\ (5,-1),~(\sqrt{23},1)\\ \text{and the sum the problem asks for is $\sqrt{23}+5$} \)
