Question

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

$x^2<10-3x$

Answer 1

let $f=x^2-10+3x$

$f=x^2+2\times \frac{3}{2}\times x+\frac{9}{4}-\frac{9}{4}-10={(x-\frac{3}{2})}^2-\frac{49}{4}$

$f<0\Rightarrow {(x-\frac{3}{2})}^2-\frac{49}{4}<0\Rightarrow -\sqrt{\frac{49}{4}}<(x-\frac{3}{2})<\sqrt{\frac{49}{4}}\Rightarrow \frac{-7}{2}<(x-\frac{3}{2})<\frac{7}{2}\Rightarrow -3.5+1.5<x<3.5+1.5\Rightarrow -2<x<5$

integers which satisfy are $-1,0,1,2,3,4,$ are $6$