The real numbers $a,b,c,d$ satisfy $a^2+b^2+c^2+1=d+\sqrt{a+b+c-d}$. Find the value of $d$.

Question

The real numbers $a,b,c,d$ satisfy $a^2+b^2+c^2+1=d+\sqrt{a+b+c-d}$. Find the value of $d$.

I've been given this question for a class I'm taking and I'm not really sure where to start. I let $\sqrt{a+b+c-d}=x$ and tried to find a factorization but haven't really gotten anywhere with it. Hints would be appreciated!

Answer 1

Hint: Solve the equation $$d^2-2d(a^2+b^2+c^2+1)+d+(a^2+b^2+c^2+1)^2-(a+b+c)=0$$ for $d$