Question

Solve the following inequality:
$\sqrt{-x^2+x+2}+2x+1>0.$

Answer 1

$\sqrt{-x^2+x+2}+2x+1>0$

For definition of square root,

$\Rightarrow -x^2+x+2\ge 0$

$\Rightarrow x^2-x-2\le 0$

$\Rightarrow x^2-2x+x-2\le 0$

$\Rightarrow (x-2)(x+1)\le 0$

Now squaring equation,

$-x^2+x+2>(-2x-1{)}^2$

$\Rightarrow -x^2+x+2>4x^2+1+4x$

$\Rightarrow 0>5x^2+3x-1$

$\Rightarrow (x-\frac{-3+\sqrt{2}9}{10})(x-\frac{-3\sqrt{2}9}{10})<0$

The range is $x\in [-1,\frac{-3+\sqrt{2}9}{10}]$