Question

$\mathrm{for \: what \: values \: of \: x \: is \: the \: expression }$
$\mathrm{x^2 - 10x +9 \: is \: positive }$

Answer 1

$\mathrm{x^2 - 10x +9 \: is \: positive }$

$\mathsf \red{step \: i : } \mathrm{find \: the \: roots \: of \: equation}$

$\mathsf{ {x}^{2} - 10x + 9 } \\ \\ \mathsf{ = {x}^{2} - 9x + x + 9 } \\ \\ \mathsf{ = x(x - 9) + (x + 9)} \\ \\ \mathsf{ = (x - 9)(x + 1)} \: \: \: \: \: \: \: \:$

$\mathsf{The \: roots \: of \: the \: equation \: are : - 1 \: and \: 9}$

$\\ \mathrm{Here \: the \: coefficient \: of \: x^2 \: is \: 1, \: which \: is \: positive} \\ \\ \mathsf \red{\: {x}^{2} - 5x - 14 \: is \: positive \: when \: \: x < - 2 \: (or) x > 7}$