Find the roots of the following quadratic equations (if they exist) by the method of completing the square.

We have to find the roots of given quadratic equation by the method of completing the square. We have,

$2 x^{2}-7 x+3=0$

We should make the coefficient of $x^{2}$ unity. So,

$x^{2}-\frac{7}{2} x+\frac{3}{2}=0$

Now shift the constant to the right hand side,

$x^{2}-\frac{7}{2} x=-\frac{3}{2}$

Now add square of half of coefficient ofon both the sides,

$x^{2}-2\left(\frac{7}{4}\right) x+\left(\frac{7}{4}\right)^{2}=-\frac{3}{2}+\left(\frac{7}{4}\right)^{2}$

We can now write it in the form of perfect square as,

$\left(x-\frac{7}{4}\right)^{2}=-\frac{3}{2}+\frac{49}{16}$

$=\frac{25}{16}$

Taking square root on both sides,

$\left(x-\frac{7}{4}\right)=\sqrt{\frac{25}{16}}$

So the required solution of,

$x=\frac{7}{4} \pm \frac{5}{4}$

$=3, \frac{1}{2}$