Question

x square +root 2x+1 is real number​

Answer 1

Answer:

We look for real solutions. Let y=2x−1. Note that x≥12. We have x=y+12. Then

x+2x−1−−−−−√=y+12+y√=12(1+y√)2.

Taking the square root, we find that

x+2x−1−−−−−√−−−−−−−−−−√=12–√(1+y√).

Almost similarly, we find that

x−2x−1−−−−−√−−−−−−−−−−√=12–√|1−y√|.

Adding, we end up with the equation

(1+y√)+|1−y√|=2–√A.

If y√≤1, we get A=2–√, with no other conditions on y. That gives us an interval of solutions when A=2–√.

If y√>1, we arrive after a little manipulation at y=A22. So there is no solution if A=1, and indeed if A<2–√. There is a unique solution if A>2–