This week, my teacher gave us this problem to solve algebraically:
sqrt(x-3) + sqrt(5) = sqrt(2) + sqrt(x + 2)
(I apologize if there's some symbols method I don't know about)
I haven't the slightest idea how I'd go about solving that. I thought maybe just isolating the radical over and over, but that doesn't seem feasible...
...but that's exactly the method you're likely expected to use. And, though it
will definitely be messy, it also
will get the job done. (As for formatting, yes, there's LaTeX coding available, but what you've used is perfectly fine.)
If you're not comfortable with the substitution method shown in the first reply, then try your method. Some might choose first to get the variable-containing terms together on one side, putting the numerical-only terms together on the other, but let's start with what they gave you, so you can see that, while awkward, this
is possible without any "tricks".
First, square both sides:
. . . . .\(\displaystyle \left(\sqrt{x\, -\, 3}\, +\, \sqrt{5}\right)^2\, =\, \left(\sqrt{2}\, +\, \sqrt{x\, +\, 2}\right)^2\)
. . . . .\(\displaystyle x\, -\, 3\, +\, 2\sqrt{5(x\, -\, 3)}\, +\, 5\, =\, 2\, +\, 2\sqrt{2(x\, +\, 2)}\, +\, x\, +\, 2\)
. . . . .\(\displaystyle x\, +\, 2\, +\, 2\sqrt{5(x\, -\, 3)}\, =\, x\, +\, 4\, +\, 2\sqrt{2(x\, +\, 2)}\)
Subtract an "x" and a "2" from either side:
. . . . .\(\displaystyle 2\sqrt{5(x\, -\, 3)}\, =\, 2\, +\, 2\sqrt{2(x\, +\, 2)}\)
Divide through by 2:
. . . . .\(\displaystyle \sqrt{5(x\, -\, 3)}\, =\, 1\, +\, \sqrt{2(x\, +\, 2)}\)
Rearrange a bit:
. . . . .\(\displaystyle \sqrt{5(x\, -\, 3)}\, -\, \sqrt{2(x\, +\, 2)}\, =\, 1\)
And now I'll bet you recognize this as a "doable" problem, though a nasty one. Two more squarings, and you should be done (other than for checking your solutions in the original equation!).
