For what integral $k$ is the inequality $x^{2} - 2 (4 k - 1) x + 15 k^{2} - 2 k - 7 > 0$ valid for any real $x$ ?

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Solution

For valid any real number D>0 which is $b^{2} - 4 a c < 0$
$a = 1$
$b = - 2 \times (4 k - 1)$
$c = 15 k^{2} - 2 k - 7$
$b^{2} - 4 a c < 0$
= $(- 2 \times (4 k - 1))^{2} - 4 \times 1 \times (15 k^{2} - 2 k - 7)$
= $4 \times (16 k^{2} - 8 k + 1) - 4 \times (15 k^{2} - 2 k - 7)$
= $4 \times (16 k^{2} - 8 k + 1 - 15 k^{2} + 2 k + 7)$
= $4 \times (k^{2} - 6 k + 8)$
= $4 (k - 2) \times (k - 4)$
= $4 \times (k - 2) \times (k - 4) < 0$
= $(k - 2) \times (k - 4) < 0$
$(k - 2) \times (k - 4) = 0, w h e n k = 2, 4$
$k^{2} - 6 k + 8 i s a l s o a n u p w a r d o p e n i n g p a r a b o l a . I t i s l e s s t h a n 0 w h e n 2 < k < 4$
$2 < k < 4$ is the answer.

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