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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.