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31a) $x^2 - x + \dfrac34$
$= x^2 - 2\cdot\dfrac12\cdot +\dfrac14 + \dfrac12$
$= \left(x -\dfrac12\right)^2 + \dfrac12$
Because $\left(x -\dfrac12\right)^2 \geq 0\quad \forall x$
So $\left(x -\dfrac12\right)^2 + \dfrac12 \geq \dfrac12>0$
Therefore $x^2 - x +\dfrac34 > 0\quad \forall x$
b) $2x^2 + x - 18$
$= 2x^2 - 6x + 7x - 21 + 3$
$= 2x(x-3) + 7(x-3) +3$
$= (x-3)(2x + 7) +3$
$2x^2 + x - 18 \quad \vdots \quad (x-3)$
$\Leftrightarrow (x-3)(2x + 7) +3 \quad \vdots \quad (x-3)$
$\Leftrightarrow 3 \quad \vdots \quad (x-3)$
$\Leftrightarrow (x-3)=\{-3;-1;1;3\}$
$\Leftrightarrow x = \{0;2;4;7\}$
32)$\quad\dfrac{a}{ab + a + 1} +\dfrac{b}{bc + b + 1} +\dfrac{c}{ca + c +1}$
$=\dfrac{ac}{abc + ac + c} +\dfrac{abc}{abc^2 + abc + ac} +\dfrac{c}{ca + c +1}$
$= \dfrac{ac}{1 + ac + c} +\dfrac{1}{c + 1+ ac} +\dfrac{c}{ca + c +1}$
$=\dfrac{ac + 1 + c}{ac + 1 + c}$
$=1$
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