Use Implicit function theorem to show F(x,y) = y^2+2xy +x-1 =0 can be solved for y in terms of x near all point (x0,y0) with $y_0\neq -x_0$
Check the result by
a) directly (using algebra)
b) determine if F(x,y)=0 is locally solvable near any point \((x_0,-x_0).
Please, help

Question

Use Implicit function theorem to show F(x,y) = y^2+2xy +x-1 =0 can be solved for y in terms of x near all point (x0,y0) with $y_0\neq -x_0$
Check the result by 
a) directly (using algebra)
b) determine if F(x,y)=0 is locally solvable near any point \((x_0,-x_0).
Please, help

loser66 · Answer

IFT says that we can solve for y in terms of x if $\dfrac{\partial F}{\partial y}\neq 0$, and we have it is $2y +2x$, hence if $y_0\neq -x_0$, it is not equal 0, hence we are done.

loser66 · Answer

For part a) I am not sure what I am supposed to do. If checking directly by using algebra, I have $y = -2x\pm\sqrt{x^2-x+1}$ But I don't know how to argue more.

loser66 · Answer

@tkhunny

tkhunny · Answer

What more is there for Part A?  Looks to me like you're done.

loser66 · Answer

I have 2 different equations, right? $\pm \sqrt....$ how to argue?

loser66 · Answer

And how to do b?

tkhunny · Answer

"$\huge{can}$ be solved for y in terms of x"

Throw one out.

Is x^2 - x + 1 < 0 anywhere?

loser66 · Answer

no solution for that

tkhunny · Answer

Then, it always exists in Real numbers and we are done with the existence and our ability to solve this one.

Substitute for $y_{0}$ as indicated.

loser66 · Answer

Thank you so much.

tkhunny · Answer

Once in a while, I get one.  :-)