Solve the system of equations

y  = x^2  - 3x + 4

x + y  =  4    ⇒   y  =  4 - x

Set the y's equal

x^2 - 3x  + 4   =  4  - x         subtract 4 from both sides

x^2  - 3x    =  - x          add x to both sides

x^2  - 2x  =   0        factor

x ( x - 2)  = 0

So.....setting each factor to 0 and solving, we have

x  = 0                   x  - 2   = 0   add 2 to both sides

                             x  = 2

So....the solutions are   x   = 0    and x  = 2