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Algebra of Complex Numbers

For what valu...

Question

For what values of $k$ is the inequality $x^{2} - (k - 3) x - k + 6 > 0$ valid for all real $x$ ?

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Solution

$x^{2} - (k - 3) x - k + 6 > 0, \forall x \in R ∴ D < 0 {(k - 3)}^{2} + 4. (k - 6) < 0 k^{2} - 6 k + 9 + 4 k - 24 < 0 k^{2} - 2 k - 15 < 0 (k - 5) (k + 3) < 0 k \in (- 3, 5)$

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