My question: Find the values of $k$ for which the parabola $y=2x^2+kx+9$ does not intersect the line $y=2x+2$.
My workings: I am thinking of using the discriminant rule to this where Δ < 0, however, I am unsure if it is applicable. As far as i know, the Δ shows the number of solutions and does the graph touches x-axis or not.
Can someone show some working outs or at least give me some hints where and how i should approach this question? Thank you very much!
Hint: Your idea of discriminant is good. Substitute for $y$ in the quadratic using the equation of the line. Now if there are solution(s) for $x$, there is intersection, so set the discriminant to negative.
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Details: We need to ensure there are no solutions for $2x+2 = 2x^2+kx+9$. $\iff 2x^2+(k-2)x+7 \neq 0 \iff (k-2)^2<4\cdot2\cdot7$ $\iff |k-2|<2\sqrt{14} \iff k \in (2-2\sqrt{14}, 2+2\sqrt{14})$.