Let $a,b,p,q,r\in\mathbb{R}$ such that $p,q>0$. I need to find or guess some $p,q,r$ such that \begin{gather} a^2-b^2+2b-1\geq0\\ \Downarrow\\ a^2\left[4(r-p)q-r^2\right]-b^2[4p^2-4pq]+2b\left(r(r-2p)a^2-\frac{ba^2r^2}{2}\right)+4pq\geq 0 \end{gather} Any ideas will be appreciated. Thanks
2026-04-24 01:07:46.1776992866
Solving quadratic inequalities $a^2-b^2+2b-1\geq0$....
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your inequality is equivalent to $$a^2-(b-1)^2\geq 0$$ using the binomial formual we obatin $$\left(a-(b-1)\right)\left(a+b-1\right)\geq 0$$ can you finish?