The ratio of roots of the equation x^2 + ax + a + 2 = 0 is 2. Find the value of a.

The equation given is x^2 + ax + a + 2 = 0 and it is told that the 'ratio' of its roots is 2. 

Now, if you think it over, you'll understand that if two numbers are in ratio 2, then one of those numbers have to be 2 times the other. Or, in other words we can say, that one of the number will be half (1/2 times) the other. Both these conditions are the same.

So, we can assume the roots of x^2 + ax + a + 2 = 0 to be t and 2t. 

(You can see it here. 2t is 2 times t, and obviously then, t is 1/2 times 2t)

So, here,

Sum of roots = -(a)/1

=> t + 2t = -a

=> 3t = -a

=> t = -a/3

Product of roots = (a + 2)/1

=> t * 2t = (a + 2)

=> 2t^2 = (a + 2)

Now, we'll substitute the value of t we found from by using sum of roots formula,

=> 2(-a/3)^2 = (a + 2)

=> 2a^2/9 = (a + 2)

=> 2a^2 = 9a + 18

=> 2a^2 - 9a - 18 = 0

This is again a Quadratic equation, and can be easily solved by splitting the middle term.

=> 2a^2 - 12a + 3a - 18 = 0

=> 2a(a - 6) + 3(a - 6) = 0

=> (2a + 3)(a - 6) = 0

=> (2a + 3) = 0; (a - 6) = 0

=> a = -3/2; a = 6.

Hence, the given condition can be satisfied by two values of a. a= -3/2, and a = 6. These values of a are the answer.