How to solve the equation 9^x -4*3^x + 3 = 0 ?

The equation is an exponential equation and we could write it in this way:

3^2x + 4*3^x + 3 = 0

It is advisable to solve this type of equation, using substitution method.

Now we can do the substitution 3^x=t

t^2 + 4*t + 3 = 0

This is a quadratic equation and to find it's roots, we can apply the quadratic formula:

t1 = [-4+sqrt(16-12)]/2

t1 = (-4+2)/2

t1 = -1

t2 = (-4-2)/2

t2 = -3

But the initial equation is not solved yet.

3^x = t1

3^x = -1, impossible because 3^x>0!

3^x = t2

3^x = -3, again impossible, because 3^x>0!

The equation has not real solutions!