3x^2+5x+4 color(white)(aaa)=color(white)(aa) 0
color(white)(aaaaaa^1a)-4color(white)(1aaa1)-4color(white)(aaa) Subtract 4 from both sides
3x^2+5xcolor(white)(aaaaaa)=-4
3(x^2+5/3xcolor(white)(aaaa))=-4color(white)(aaa)Factor out the leading coefficient 3
Divide the coefficient of the x term by 2: color(white)(aa)5/3-:2=5/6
Square the result: (5/6)^2=25/36 and add it to both sides:
3(x^2+5/3x+25/36)=-4+3(25/36)color(white)(aa)Note: add 3*25/36 on thecolor(white)(aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa) right to account for the color(white)(aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa)3 factored out on the left.
3(x+5/6)(x+5/6)=-23/12color(white)(aaa) Factor the left side and color(white)(aaaaaaaaaaaaaaaaaaaaaaaaaaaa) simplify the right
Note the * 5/6 in the binomial (x+5/6)is the same as the coefficient of the x term divided by 2 above.*
(x+5/6)(x+5/6)=(x+5/6)^2...thus...
3(x+5/6)^2=-23/12
3/3(x+5/6)^2=(-23/12)/3color(white)(aaa)Divide both sides by 3
(x+5/6)^2=-23/36
sqrt((x+5/6)^2)=sqrt(-23/36)color(white)(aaa)Square root both sides
x+5/6=+-sqrt23/6i
Subtracting 5/6 from both side gives...
x=-5/6+-sqrt23/6i
