State whether the following quadratic equations have two distinct real roots. Justify your answer.
$ (x+1)(x-2)+x=0 $

Given:

\( (x+1)(x-2)+x=0 \)

To do:

We have to state whether the given quadratic equations have two distinct real roots.

Solution:

$(x+1)(x-2)+x=0$

$x^{2}+x-2 x-2+x=0$

$x^{2}-2=0$

Comparing with $a x^{2}+b x+c=0$, we get,

$a=1, b=0$ and $c=-2$

Therefore,

Discriminant $D=b^{2}-4 a c$

$=(0)^{2}-4(1)(-2)$

$=0+8$

$=8>0$

Hence, the equation $(x+1)(x-2)+x=0$ has two distinct real roots.