Question

Find the roots of the quadratic equation (if they exist) by the method of completing the square.

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

Answer 1

$2x^2+x+4=0$ can be written as $x^2+\frac{x}{2}+2=0$

$\Rightarrow x^2+\frac{x}{2}=-2$

Add the coefficient of ${\left(\frac{x}{2}\right)}^2$ to both sides

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

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

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

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

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

Since root cannot be negative

Therefore, no real roots exist.