Question

by using the completing square method find the roots of the equation 4x square +4x+3

Answer 1

4x^2+4x+3=0

=> x^2+x+3/4=0

=> x^2+x=-3/4
add (1/2)^2 both side

=> x^2+2.x (1/2) + (1/2)^2 =(-3/4)+(1/2)^2

=>(x+1/2)^2=(-2/4)

here you see left side is square term means always positive but in right side is negative term so, this quadratic equation have no real roots .
here only imaginary roots possible .
hence,
(x+1/2)^2 =(-1/2)
take square root both side
x+1/2 =+_i/root2
x= -1/2 +_i/root2

Answer 2

=> 4x^2 +4x +3 =0
=> 4x^2 +4x = -3
=> x^2 +x = -3/4 (dividing the whole equation by 4)

=> x^2 +x +(1/2)^2 = -3/4 +(1/2)^2
=> (x +1/2)^2 = -3/4 +1/4
=> (x +1/2)^2 = -2/4

The equation will have no real roots.