+0

# Help with 3 problems

0
172
1

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. Aug 5, 2019 ### 1+0 Answers #1 +1 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)

$$|y| - \dfrac{2x}{|x|} = -1\\ |y|-2\text{sgn(x)= -1}\\ |y|=-1 + 2\text{sgn(x)}\\ \text{The only solutions to this are }\\ x>0,~y = \pm 1$$

$$\text{applying the results of the previous equation to the second}\\ x^2 +1 = 24\\ x^2 -1 = 24\\ x =\sqrt{23}, ~ 5\\ \text{the ordered pairs of solutions are }\\ (5,-1),~(\sqrt{23},1)\\ \text{and the sum the problem asks for is \sqrt{23}+5}$$

.
Aug 6, 2019