State whether the following quadratic equations have two distinct real roots. Justify your answer.
\( 2 x^{2}-6 x+\frac{9}{2}=0 \)

Given:

\( 2 x^{2}-6 x+\frac{9}{2}=0 \)

To do:

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

Solution:

$2 x^{2}-6 x+\frac{9}{2}=0$

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

$a =2, b=-6$ and $c=\frac{9}{2}$

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

$=(-6)^{2}-4(2)(\frac{9}{2})$

$=36-36$

$=0$

$D=0$

Hence, the equation $2 x^{2}-6 x+\frac{9}{2}=0$ has equal and real roots.