How many roots are there of the following equation? Find them.

3

^{2x+2}− 3^{x+3}− 3^{x}+ 3 = 03^{2x+2} − 3^{x+3} – 3^{x} + 3 = 0. Let us try grouping:

3

3

(3

^{2x+2}− 3^{x+3}− 3^{x}+ 3 = 03

^{x}· 3^{x + 2}− 3^{1}· 3^{x+2}− (3^{x}− 3) = 0(3

^{x}− 3)(3^{x+2}) − 1(3^{x}− 3) = 0So we’ve factored it: (3^{x+2} − 1)(3^{x} − 3) = 0

→ (9·3^{x} − 1)(3^{x} − 3) = 0

9·3

3

^{x}− 1 = 0 → 3^{x}= 1/9 →*x*= -23

^{x}− 3 = 0 → 3^{x}= 3 →*x*= 1. We’ve got two roots.Check:

*x*= -2 : 3

^{-4+2}− 3

^{-2+3}− 3

^{-2}+ 3 = 0 ✓

*x*= 1 : 3

^{4}− 3

^{4}− 3

^{1}+ 3 = 0 ✓