For what value of b will this equation have only one real root: (2b−5)x^2−2(b−1)x+3=0.

The discriminant of the quadratic formula must = 0 to have only one root

b^2 - 4ac = 0

b in the quadratic formula for this case is -2b+2 a is 2b-5 c is 3

Plug these values into the red equation to find b = 4