Question

The value of \( x \) such that \( 3^{2x} - 2(3^{x+2}) + 81 = 0 \) is:

• A. \( 1 \)
• B. \( 2 \)
• C. \( 3 \)
• D. \( 4 \)
• E. \( 5 \)

Hint:

Always look for a common base in exponential terms. If you see $a^{2x}$ and $a^x$, it's a strong signal that you should use a quadratic substitution.

Answer 1

The Correct Option is B

Concept: Exponential equations can often be transformed into quadratic equations using a substitution. Here, we can let \( y = 3^x \).

Step 1: Rewrite the equation.
Using the property \( 3^{x+2} = 3^x \cdot 3^2 = 9 \cdot 3^x \): \[ (3^x)^2 - 2(9 \cdot 3^x) + 81 = 0 \] \[ (3^x)^2 - 18(3^x) + 81 = 0 \]

Step 2: Solve the quadratic equation.
Let \( y = 3^x \). The equation becomes: \[ y^2 - 18y + 81 = 0 \] This is a perfect square trinomial: \[ (y - 9)^2 = 0 \quad \Rightarrow \quad y = 9 \]

Step 3: Solve for \( x \).
Substitute back \( y = 3^x \): \[ 3^x = 9 \] \[ 3^x = 3^2 \quad \Rightarrow \quad x = 2 \]