[2at - t^2b] / 2 = c multiply both sides by 2
2at - t^2b = 2c multiply throgh by -1
bt^2 - 2at = -2c we will complete the square on t.......divide both sides by b
t^2 - (2a/b)*t = [-2c] /b take 1/2 the coefficient on t = a/b .....square it and add to both sides
t^2 - (2a/b)*t + (a/b)^2 = (a/b)^2 - [2c]/b factor the left side
[ t - (a/b) ]^2 = (a/b)^2 - [2c]/b take the pos/neg roots of both sides
t - (a/b) = ± √ [ (a/b)^2 - [2c]/b] add (a/b) to both sides
t = ± √ [ (a/b)^2 - [2c]/b] + (a/b) we can clean this up a little
t = ± √ [ a^2 - 2bc] / b + a/b
t = ( a ± √ [ a^2 - 2bc] ) / b [ we assume that b is not 0 ]