Solution — 2017 Question 1 Quadratic Equations & Inequalities

For the quadratic to be always positive, the graph must lie entirely above the $x$ -axis, which means it must have no real roots. Hence the discriminant must be negative.

\begin{matrix} b^{2} - 4 a c & < 0 \\ {(k - 4)}^{2} - 4 (- (k - 7)) (1) & < 0 \\ k^{2} - 8 k + 16 + 4 k - 28 & < 0 \\ k^{2} - 4 k - 12 & < 0 \\ (k + 2) (k - 6) & < 0 \end{matrix}

$- 2 < k < 6 ∎$