Find the roots of the following quadratic equations, if they exist, by the method of completing the square:
$2x^2 + x + 4 = 0$

Given:

Given quadratic equation is $2x^2+x +4 = 0$.

To do:

We have to find the roots of the given quadratic equation.

Solution:

$2x^2+x + 4 = 0$

$2(x^2 + \frac{1}{2} x +\frac{4}{2}) = 0$  

$x^2 + \frac{1}{2} x + 2 = 0$  

$x^2 + 2\times \frac{1}{2}\times \frac{1}{2} x = -2$  

$x^2 + 2\times \frac{1}{4} x = -2$  

Adding $(\frac{1}{4})^2$ on both sides completes the square. Therefore,

$x^2 + 2\times (\frac{1}{4}) x + (\frac{1}{4})^2 = -2+(\frac{1}{4})^2$

$(x+\frac{1}{4})^2=-2+\frac{1}{16}$      (Since $(a+b)^2=a^2+2ab+b^2$)

$(x+\frac{1}{4})^2=\frac{1-2\times16}{16}$

$(x+\frac{1}{4})^2=\frac{1-32}{16}$

$(x+\frac{1}{4})^2=\frac{-31}{16}$

$x+\frac{1}{4}=\pm \sqrt{\frac{-31}{16}}$     (Taking square root on both sides)

$x=\sqrt{\frac{-31}{16}}-\frac{1}{4}$ or $x=-\sqrt{\frac{-31}{16}}-\frac{1}{4}$

$x=\frac{\sqrt{-31}-1}{4}$ or $x=-(\frac{\sqrt{-31}+1}{4})$

Therefore, no real roots exist for the given quadratic equation.