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-

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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