How do I approach the following problem. Please don't give answer but just guidance as to how to approach these in general. I need to be able to solve without a calculator.

3^(2x+1)-16(3^x)+5=0

Distribute

Eventually answered my own question. No idea what distributing has to do with anything though...

I simplified 3^(2x+1) to ((3^x)^2)*3^x then substituted a variable for 3^x and solved the corresponding quadratic. Seems to have worked

I think you were on the right track, except

3^(2x+1) = 3(3^x)(3^x) = 3(3^x)^2

so let y = 3^x to change your equation to
3y^2 - 16y + 5 = 0
(y - 5)(3y - 1) = 0
y = 5 or y = 1/3

so 3^x = 5 or 3^x = 1/3

This cannot be solved WITHOUT a calculator

3^x = 5
log 3^x = log 5
x log3 = log 5
x = log5/log3 = appr. 1.465

but 3^x = 1/3
3^x = 3^-1
x = -1

so x = -1 or x = appr. 1.465

both answers verify in the original

To approach this problem without a calculator, you can use a method called substitution. The idea is to replace the variable with a new variable that will simplify the equation.

1. Start by letting y = 3^x. This substitution will help us simplify the equation.

2. Rewrite the equation using the new variable.
The original equation becomes:
y^2 - 16y + 5 = 0

3. Solve the quadratic equation. You can use factoring, completing the square, or the quadratic formula to find the values of y.

4. Substitute back the original variable. Once you have found the values of y, you can substitute them back into the equation:
For each value of y, solve the equation 3^x = y to find the corresponding value(s) of x.

By following these steps, you can solve the given equation without using a calculator. Remember to show your work and carefully solve each step of the process.