Solve the following quadratic equations by the method of completing the square:

2x² + x + 4 = 0

Now in the above quadratic equation the coefficient of x² is 2. Let us make it unity by dividing the entire quadratic equation by 2.

x² + 1/2 x + 2= 0

x² + 1/2 x = - 2

Now by taking half of the coefficient of x and then squaring it and adding on both LHS and RHS sides.

Coefficient of x = 1/2

Half of 1/2 = 1/4

Squaring the half of 1/2 = 1/16

Now the LHS term is a perfect square and can be expressed in the form of (a-b) ² = a² – 2ab + b² where a = x and b = 1/4

On simplifying both RHS and LHS we get an equation of following form,

(x ± A)² = k²

It is observed that the term obtained on RHS is a negative term and taking square root of a negative term will give imaginary roots for the given quadratic equation.

Therefore the given quadratic equation does not has real roots.