Two numbers `ba n dc` are chosen at random with replacement from the numbers 1, 2, 3, 4, 5, 6, 7, 8 and 9. Find the probability that `x2+b x+c >0` for all `x in R` .

Explanation

We have
`a^3+b^3+c^3-3abc`
`=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)`
`=(1)/(2)(a+b+c)xx [2a^2+2b^2+2c^2-2ab-2bc -2ca]`
` (1)/(2) (a+b+c)xx [(a^2-2ab+b^2)+(b^2-2bc+c^2)+(c^2-2ca+a^2]`
`=(1)/(2)(a+b+c)xx[(a-b)^2+(b-c)^2+(c-a)^2]`.
`therefore a^3+b^3+c^3-3abc =(1)/(2)(a+b+c)xx (a-b)^2+(b-c)^2+(c-a)^2]`.