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At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

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Find how many integers \(a\) are there such that \(\large \color{black}{\begin{align} 9x^2+3ax+a+5>0,\ x\in \mathbb{R} \hspace{.33em}\\~\\ \end{align}}\) for all values of \(x\). \(\large \color{black}{\begin{align} &a.) \ 7\hspace{.33em}\\~\\ &b.) \ 8\hspace{.33em}\\~\\ &c.) \ 9\hspace{.33em}\\~\\ &d.) \ 10\hspace{.33em}\\~\\ \end{align}}\)
i think you need to solve \[a^2-4a-20<0\] for that parabola to be >0 since the leading coefficient is >0 must be above x-axis so for that to be true the discriminant must be less than zero
so we need to draw the parabola a^2-4a-20 to see the range of integers where it is <0

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according to this picture here https://www.desmos.com/calculator the range for a such that a^2-4a-20<0 -2.9
looks like 9 to me
took me to time to realize this!
but i might be wrong for my skills lol
excellent!

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