Question

By using the method of completing the square, show that the equation $2x^2+x+4=0$ has no real roots.

Answer 1

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

a=2 b=1 c=4

If the quadratic equation has no real roots
$b^2-4ac$ < 0
1- $4\times 2\times 4$ <0
1-32<0
-31<0
hence it's prove that equation has no real roots.

$2x^2+x$ = -4
$x^2+\frac{x}{2}+\frac{1}{4}$ = -2+$\frac{1}{4}$
$(x+\frac{1}{2}{)}^2$ = $\frac{-7}{4}$
(x + $\frac{1}{2}$) = ±$\frac{\sqrt{-}7}{2}$
x = $\frac{\sqrt{-}7}{2}$-$\frac{1}{2}$=√-7-$\frac{1}{2}$

as there is negative sign present in the root, the roots are not real.