X^2 +(2a-1)x+b=0 find the relation between a and b if root of x is a-1

Question

Quadratic equation 

Answer 1

If x²+(2a-1)x+b=0 and x₁=a-1, x1+x2 = 1-2a, x1*x2=b , x1=a-1 , x2 = 1-2a-(a-1) = 2-3a

then [x-(a-1)]*[x-(2-3a)] = x² -x{(a-1)+(2-3a)} + (a-1)(2-3a) = x² - x * (1-2a) +b

So b = (a-1)(2-3a)

If x1=x2=a-1, then a-1=2-3a , 4a=3, a=3/4 , b=(a-1)(2-3a)=(-1/4)*(-1/4)=1/16

To validate the solution, put the root of x back

(x-(a-1))=(x+1/4)

(x+1/4)² = x²+1/2*x+1/16

2a-1 = 2*3/4 -1 =1/2