what is the solution set of the equation 4^2-9=0

Is something missing from your equation?

As it is...

4² - 9   =   16 - 9   =   7     ≠   0      It is just a false equation.

I think this might be

4x^2  -  9  = 0        add 9 to both sides

4x^2  =   9            divide both sides by 4

x^2  =   9 / 4          take both roots

x  = ± √ [9 / 4]  =  ± 3 / 2