Question

Find the solution set of 3^2(x+1) - 8.3^x+1 = 9

Answer 1

3^2(x+1) - 8 * 3^x+1 = 9
3^2(x+1) - 8 * 3^x+1 -9 =0
This is a quadratic equation in 3^x+1
So let 3^x+1 = t
We can now write the equation as:
t² - 8 t -9=0
t² +t - 9t -9=0
t ( t+1) -9 (t+1) =0
(t-9) (t+1) = 0
So t=9 OR t= -1
But we have t= 3^x+1
We know that 3^x+1 can never be <0; It is always >0 for all values of 'x';
so t=-1 is not possible;
So we are left with t=9 or
3^x+1 = 9 = 3²
since bases are equal, we can equate the powers to get:
3^x+1 = 3²
x+1 = 2
x= 2-1=1;

Thus, the solution set of this equation is x= {1}