Question

Given: \( 3^{2x} - 12 \times 3^{x} + 27 = 0 \).
Quantity A: x
Quantity B: \(3^{x}\)

• A. Quantity A is greater.
• B. Quantity B is greater.
• C. The two quantities are equal.
• D. The relationship cannot be determined from the information given.
• E.

Hint:

In equations involving powers, substitution can simplify complex expressions.

Answer 1

The Correct Option is D

We are given the equation: \[ 3^{2x} - 12 \times 3^{x} + 27 = 0 \] Step 1: Let \( y = 3^x \). The equation becomes: \[ y^2 - 12y + 27 = 0 \] Step 2: Solve this quadratic equation using the quadratic formula: \[ y = \frac{-(-12) \pm \sqrt{(-12)^2 - 4(1)(27)}}{2(1)} = \frac{12 \pm \sqrt{144 - 108}}{2} = \frac{12 \pm \sqrt{36}}{2} \] \[ y = \frac{12 \pm 6}{2} \] So, \( y = 9 \) or \( y = 3 \). Step 3: Since \( y = 3^x \), we have: - If \( y = 9 \), then \( 3^x = 9 \), so \( x = 2 \). - If \( y = 3 \), then \( 3^x = 3 \), so \( x = 1 \). Step 4: Thus, we have two possible values for \( x \) (1 and 2). As we don't know which one is correct from the given information, we cannot definitively compare \( x \) with \( 3^x \) without additional context.