Mathematico | Mathematico

Find the sum of all real solutions of the equation

3^{2 x} - 4 \times 3^{x + 1} + 27 = 0

Hint

This can be written as

{(3^{x})}^{2} - 4 \times 3 \times 3^{x} + 27 = 0

Hint

Let $u = 3^{x}$ and solve

u^{2} - 12 u + 27

Solution

The equation can be written as

{(3^{x})}^{2} - 4 \times 3 \times 3^{x} + 27 = 0

Now let $u = 3^{x}$ to get

\begin{aligned} u^{2} - 12 u + 27 & = 0 \\ (u - 9) (u - 3) & = 0 \end{aligned}

The solutions are

\begin{aligned} 3^{x} & = 9 \\ x & = 2 \end{aligned}

and

\begin{aligned} 3^{x} & = 3 \\ x & = 1 \end{aligned}