<s>[INST] How many real roots for $ax^2 + 12x + c = 0$? If $a$ and $c$ are integers and  $2 < a < 8$ and  $-1 < c$, how many equations of the form $$ax^2+12x+c=0$$ have real roots?
 [/INST] As @DonAntonio give hint a quadratic eqn have real roots iff
$$b^2-4ac\ge0$$
$$144-4ac\ge0\implies ac\le36$$
since $2<a<8$ and $c>-1$ so $a$ will be in set {$3,4,5,6,7$} and $c$ {$0,1,2,3,\cdots$}
from $$ac\le36$$
firstly we will check combination for equality so factors of $36$ in given range
$36=3\times 12,4\times 9,6\times 6$ 
so possible combination of a and c will :
for $a=3,$ $c$ will {$0,1,2,3,4,5,6,7,8,9,10,11,12$}
for $a=4,$ $c$ will {$0,1,2,3,4,5,6,7,8,9$}
for $a=5,$ $c$ will {$0,1,2,3,4,5,6,7$} (this will hold only "<" relation since $5$ is not divider of $36$)
for $a=6,$ $c$ will {$0,1,2,3,4,5,6$}
for $a=7,$ $c$ will {$0,1,2,3,4,5$} (same case as $a=5$)
so possible eqn are  $13+10+8+7+6=44$
$44$ equation having a and c in given range have real roots
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