Given equation 9^{x + 2} - 6.3^{x + 1 }+ 1 = 0
⇒ 9^{x}.9^{2 }– 6.3^{x}.3^{1} + 1 = 0
⇒ 81.(3^{2})^{x }- 18.3^{x }+ 1 = 0
⇒ 81.3^{2x} – 18.3^{x} + 1 = 0
Putting 3^{x} = y, then it becomes 81y^{2} – 18y + 1 = 0
⇒ 81y^{2} – 9y – 9y + 1 = 0
⇒ 9y(9y – 1) – 1(9y – 1) = 0
⇒ (9y- 1)(9y – 1) = 0
⇒ 9y = 1 ⇒ y = 1/9
But 3^{x} = 1/9 = 1/3^{2} = 3^{-2}
∴ x = - 2
Hence, the required root is – 2.