Question

if a-d, a, a+d are the zeroes of 2x3-15x2+37x-30.find a and d

Answer 1

As the zeroes are a-d, a, a+d. they are in A.P.  (not matters here!)
The given polynomial 2x³-15x²+37x-30.

Sum of it's roots
a-d+a+a+d = - coefficient of x²/coefficient of x³ 
3a = 15/2
a = 5/2


sum of the product of the roots
(a-d)(a)+(a-d)(a+d)+(a+d)(a) = coefficient of x/coefficient of x³ 
⇒ a²-ad+a²-d²+a²+ad = 37/2
⇒ 3a²-d² = 37/2
⇒ 3(5/2)²-d² = 37/2
⇒ 75/4 - d² = 37/2
⇒ d² = 75/4 - 37/2
⇒ d² = 1/4
So, d = 1/2

Therefore, zeroes
a-d = 5/2-1/2 = 2
a = 5/2
a+d = 5/2+1/2 = 3

where a=5/2, d=1/2