State whether the following quadratic equations have two distinct real roots. Justify your answer.
$ \sqrt{2} x^{2}-\frac{3}{\sqrt{2}} x+\frac{1}{\sqrt{2}}=0 $

Given:

\( \sqrt{2} x^{2}-\frac{3}{\sqrt{2}} x+\frac{1}{\sqrt{2}}=0 \)

To do:

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

Solution:

\( \sqrt{2} x^{2}-\frac{3}{\sqrt{2}} x+\frac{1}{\sqrt{2}}=0 \)

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

$a=\sqrt{2}, b=-\frac{3}{\sqrt{2}}$ and $c=\frac{1}{\sqrt{2}}$

Therefore,

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

$=(-\frac{3}{\sqrt{2})^{2}-4 \sqrt{2}(\frac{1}{\sqrt{2}})$

$=\frac{9}{2}-4$

$=\frac{9-8}{2}$

$=\frac{1}{2}>0$

$D>0$

Hence, the equation \( \sqrt{2} x^{2}-\frac{3}{\sqrt{2}} x+\frac{1}{\sqrt{2}}=0 \) has two distinct real roots.