maths

Question

Each coefficient in the equation ax^2+bx+c=0 is determined by throwing an ordinary six faced die.Find the probability that the equation will hav real roots

Comments

To have real roots we need to have √((b^2)-4ac)>0.
Now a,b,c are three variable who can take values from[1-6]. Now find the combination in which above equation statify.

Answer 1

An equation ax^2 + bx + c = 0 has real roots when b^2 is greater than 4*a*c
Total no of ways to choose coefficients = 6^3 = 216.

when b=1 there are no roots
when b=2 there are roots only when a=1 and c=1
when b =3 (1,1), (2,1) and (1,2) satisfies condition, so 3 ways
when b= 4 (1,1), (1,2) (2,1) (2, 2) (1,3) (3,1) (1,4) (4,1) satisfies, so 8 ways
when b=5 (1,1),(1,2), (2,1), (1, 3), (3,1), (1,4), (4,1), (1,5),(5,1) (1,6) (6,1), (2,3), (3,2), (2,2), so 14 ways
when b=6 (1,1),(1,2), (2,1), (1,3), (3,1), (1,4), (4,1), (1,5),(5,1) (1,6) (6,1), (2,3), (3,2), (2,2) (2,4), (4,2),(3,3), so 17 ways

total of 43 ways

Probability = 43/216