Given that the curve y=2x^2 −19x+18 does not intersect the line y=x+k, find the largest integer of k.


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Question Number 120705 by ZiYangLee last updated on 02/Nov/20

$Given that the curve y = 2 x^{2} - 19 x + 18$ $does not intersect the line y = x + k,$ $find the largest integer of k .$
Commented by john santu last updated on 02/Nov/20

$\Rightarrow 2 x^{2} - 19 x + 18 = x + k$ $\Rightarrow 2 x^{2} - 20 x + 18 - k = 0$ $t a k e d i s c r i m i n a n t < 0$ $\Rightarrow 400 - 4.2 . (18 - k) < 0$ $\Rightarrow 256 + 8 k < 0$ $\Rightarrow 32 + k < 0, k < - 32 s o w e g e t k = - 33$ $t h e l a r g e s t i n t e g e r o f k$
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