When is 2 x - 3 equal to x squared minus 5

Solve: 2x - 3 = x^2 - 5 (at least, this is my interpretation of your 2 x - 3).

x^2 - 5 - 2x + 3 + 0, or x^2 - 2x - 2 = 0

I will solve this by "completing the square:"

x^2 - 2x + (1) ^2 - (1) ^2 - 2 = 0, or x^2 - 2x + 1 - 1 - 2 = 0

This is (x-1) ^2 = 3. Taking the sqrt of both sides, we get:

x-1 = plus or minus sqrt (3). Then x=1+sqrt (3) or x=1-sqrt (3). These are the two values of x for which 2x - 3 = x^2 - 5 is true.