Which of the following expressions are positive for all real values of x? a) x^2 − 2x + 5 b) x^2 −2x−1 c) x^2 +4x+2 d) 2x^2 −6x + 5


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Question Number 37940 by Rio Mike last updated on 19/Jun/18

$W h i c h o f t h e f o l l o w i n g$ $e x p r e s s i o n s a r e p o s i t i v e f o r$ $a l l r e a l v a l u e s o f x ?$ $a) x^{2} - 2 x + 5 b) x^{2} - 2 x - 1$ $c) x^{2} + 4 x + 2 d) 2 x^{2} - 6 x + 5$
	Answered by MrW3 last updated on 19/Jun/18

	$a) x^{2} - 2 x + 5 = x^{2} - 2 x + 1 + 4 = {(x - 1)}^{2} + 4 ⩾ 4 > 0$ $b) x^{2} - 2 x - 1 = x^{2} - 2 x + 1 - 2 = {(x - 1)}^{2} - 3 ⩾ - 3$ $c) x^{2} + 4 x + 2 = x^{2} + 4 x + 4 - 2 = {(x + 2)}^{2} - 2 ⩾ - 2$ $d) 2 x^{2} - 6 x + 5 = 2 (x^{2} - 3 x + \frac{9}{4}) + \frac{1}{2} = 2 {(x - \frac{3}{2})}^{2} + \frac{1}{2} ⩾ \frac{1}{2} > 0$ $\Rightarrow a n s w e r i s a) a n d d)$
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