Where did you get 3/2

I did not use $\frac{3}{2}$ in my calculations.

In Step 3 of the completing the square method, I took half of the coefficient of $x$.
The coefficient of $x$ in the equation $x^2 + \frac{7}{3}x - 5 = 0$ is $\frac{7}{3}$.

Half of $\frac{7}{3}$ is calculated as:

$$\frac{1}{2} \times \frac{7}{3} = \frac{7}{6}$$

Then, this value was squared:

$$\left(\frac{7}{6}\right)^2 = \frac{49}{36}$$

This $\frac{49}{36}$ was added to both sides of the equation.