Question

Two numbers $b$ and $c$ are chosen at random (with replacement) from the numbers $1,2,3,4,5,6,7,8$ and $9$. The probability that $x^2+bx+c>0$ for all $x\in R$ is $\frac{4k}{81}$. Find the value of $k$ ?

Answer 1

if $x^2+bx+c>0$
then $b^2-4c<0$
Tabulating possible values of b,c satisfying the condition

b	c such that $b^2<4c$	Total no. of values
1	1-9	9
2	2-9	8
3	3-9	7
4	5-9	5
5	7-9	3
6	-	0
7	-	0
8	-	0
9	-	0
	Total	32

Total possible values of b,c = $9\times 9=81$
$P=\frac{32}{81}$