Question

Number of integers in range of $x$ which satisfies the following

$x^2+x<12$

Answer 1

Given $f=x^2+x-12$ and $f<0$

$x^2+2\times \frac{1}{2}\times x+{\left(\frac{1}{2}\right)}^2-{\left(\frac{1}{2}\right)}^2-12\Rightarrow {(x-\frac{1}{2})}^2-\left(\frac{1}{4}\right)-12\Rightarrow {(x-\frac{1}{2})}^2-\left(\frac{49}{4}\right)<0\Rightarrow {(x-\frac{1}{2})}^2<\left(\frac{49}{4}\right)\Rightarrow \left(\frac{-7}{2}\right)<(x-\frac{1}{2})<\left(\frac{7}{2}\right)\Rightarrow \left(\frac{-7}{2}\right)<(x-\frac{1}{2})<\left(\frac{7}{2}\right)\Rightarrow -3.5+0.5<x<3.5+0.5\Rightarrow -3<x<4$

so possible integers are $-2,-1,0,1,2,3$

Total of 6 integers.