Question

The question

Answer 1

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 \(x\).

\(\large \color{black}{\begin{align}
&a.) \ 7\hspace{.33em}\\~\\
&b.) \ 8\hspace{.33em}\\~\\
&c.) \ 9\hspace{.33em}\\~\\
&d.) \ 10\hspace{.33em}\\~\\
\end{align}}\)

Answer 2

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

Answer 3

so we need to draw the parabola a^2-4a-20 to see the range of integers where it is <0

Answer 4

according to this picture here
https://www.desmos.com/calculator
the range for a such that a^2-4a-20<0
-2.9<a<6.9
how many integers are there in that range

Answer 5

looks like 9 to me

Answer 6

took me to time to realize this!

Answer 7

but i might be wrong for my skills lol

Answer 8

@freckles

Answer 9

excellent!