## mathmath333 one year ago The question

1. mathmath333

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}}

2. xapproachesinfinity

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

3. xapproachesinfinity

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

4. xapproachesinfinity

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

5. xapproachesinfinity

looks like 9 to me

6. xapproachesinfinity

took me to time to realize this!

7. xapproachesinfinity

but i might be wrong for my skills lol

8. xapproachesinfinity

@freckles

9. mathmath333

excellent!