Thread: Probability of at least one real root

Hi, if a quadratic equation is of the form:
x^2+sqrt(p)*x+q=0 where p can be any value in range [0,a] and q in the range [-b,b] then what is the probabillty that at least one root of the equation is real?
E.g:
if a = 4 and b = 2 ,then
probability is 0.625 and for a = 1 and b =2
probability = 0.53125
Thanks.

Hint


The equation has at least a real root iff p - 4 q >= 0 . The asked probability is the quotient of two adequate areas in the pq plane.

Originally Posted by FernandoRevilla

Hint


The equation has at least a real root iff p - 4 q >= 0 . The asked probability is the quotient of two adequate areas in the pq plane.

Thanks but i couldn't get what's quotient of two adequate areas in the pq plane?

Originally Posted by pranay

Thanks but i couldn't get what's quotient of two adequate areas in the pq plane?

For example, for a = 4 , b = 2 the region of the pq plane satisfying

0 <= p <= 2 and - 4 <= q <= 4 is a square ( S )


The region satisfying

0 <= p <= 2 and - 4 <= q <= 4 and p >= 4q is a trapezium ( T )

Then, the probability is

P = ( area of T ) / (area of S )