Each coefficient in the equation ax2 + bx + c = 0 is determined by throwing an ordinary die. Find the probability that equation will have real roots.
Answers
P = 119/(36^3)(27)
Answer:
43/216
Step-by-step explanation:
Throwing an die
6 * 6 * 6 = 216
b² - 4ac ≥ 0 for real roots
if b = 6
=> 36 - 4ac ≥ 0
=> ac ≤ 9
( a = 1 , c = 1 , 2 , 3 , 4 , 5 , 6)
a = 2 , c= 1 , 2 , 3 , 4
a= 3 , c = 1 , 2 , 3
a = 4 , c = 1, 2
a = 5 , c = 1
a = 6 , c = 1)
17 cases
if b = 5
=> 25 - 4ac ≥ 0
=> ac ≤ 6
a = 1 , c = 1 , 2 , 3 , 4 , 5 , 6
a = 2 , c= 1 , 2 , 3
a= 3 , c = 1 , 2
a = 4 , c = 1
a = 5 , c = 1
a = 6 , c = 1
14 cases
if b = 4
=>16 - 4ac ≥ 0
=> ac ≤ 4
a = 1 , c = 1 , 2 , 3 , 4
a = 2 , c= 1 , 2
a= 3 , c = 1
a = 4 , c = 1
8 cases
if b = 3
=>9 - 4ac ≥ 0
=> ac ≤ 2
a = 1 , c = 1 , 2
a = 2 , c= 1
3 cases
if b = 2
=>4 - 4ac ≥ 0
=> ac ≤ 1
a = 1 , c = 1
1 case
no case for b = 1
17 + 14 + 8 + 3 + 1 + 0 = 43 cases
probability that equation will have real roots = 43/216