Answer:

FIRST let's talk about real ROOTS.

Since each of the coefficients a, b, and c can take the values from 1 to 6

Therefore,total number of equations = 6 6 6 = 216.

The roots of the equation ax2 + bx + c = 0 will be real, if b2 - 4 ac 0 b2 4 ac.

The favourable number of elementary events can be enumerated as under:

ac a c 4ac b(so that b2 4ac) Number of ways

1 1 1 4 2,3,4,5,6 1 5 = 5

2 1,2 2,1 8 3,4,5,6 2 4 = 8

3 1,3 3,1 12 4,5,6 2 3 = 6

4 1,4,2 4,1,2 16 4,5,6 3 3 = 9

5 1,5 5,1 20 5,6 2 2 = 4

6 6,2,3 1,3,2 24 5,6 4 2 =8

7 ac is not possible 0

8 2,4 4,2 32 6 2 1 = 2

9 3 3 36 6 1

Total = 43

Since b2 4 ac and the maximum value of b2 is 36, therefore ac = 10 , 11,12 ... etc. is not possible.Therefore,we reject such values.

Therefore, Total number of favourable elementary events = 43.

Hence, REQUIRED probability= 43/216

Now, Real and equal ⇒b² −4ac=0

Each coefficients is an integer, So we consider the following cases

b=1⇒ 41 =ac No integral value of a and c

b=2⇒1=ac∴(1,1)

b=3⇒ 29 =ac No integral value of a and c

b=4⇒4=ac∴(1,4),(2,2),(4,1)

b=5⇒ 225 =ac No integral value of a and c

b=6⇒9=ac∴(3,3)

Thus we have 5 favorable ways for b=2,4,6

Total number of equation is 6.6.6=216

Therefore required probability is

5/216

Now Real and Unequal.. its simple Real - Equal

So 43/216 - 5 /216 = 38/216

For not real, subtract total possibilities from real ones.

216-43 = 173/216

I have tried my best and hope it's correct..

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Step-by-step explanation:

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