#0.4t^2 + 0.7t = 0.3t -0.2#
Subtract #0.3t# from both sides
#0.4t^2 +0.4t = -0.2#
Divide the equation by the leading coefficient of the #t^2# term #0.4#.
#t^2 + t = -0.5#
To complete the square, divide the coefficient of the #t# term by #2#.
#t^2 + color(red)1t = -0.5#
#color(red)1 / 2 = color(blue)(0.5#
Square #color(blue)(0.5)# and add it to both sides of the equation.
#t^2 +t +(color(blue)(0.5))^2=-0.5 +(color(blue)(0.5))^2#
Simplify
#t^2 +t + 0.25 = -0.25#
Factor into a binomial squared. Note that the second term of the binomial is #color (blue)(0.25)#.
#(t+color(blue)(0.5))^2 = -0.25#
Square root both sides. Note that the negative sign in front of #-0.25# results in an #i#
#sqrt((t+color(blue)(0.5))^2) = sqrt(-0.25)#
#t + 0.5 = +-0.5i#
Subtract #0.5# from both sides. Place it in front of the #+-#.
#t=-0.5+-0.5i#
