<p>Find the number of real values of <i>x</i> satisfying the equation \(9^{2\log_9 x} + 4x + 3 = 0\).</p>
Step-by-Step Solution
Key Concept: Recognize that exponential expressions are always positive; if the resulting equation yields only negative solutions, there are no real values of x.
<p><strong>Solution:</strong> Let $y = \log_9 x$, so $x = 9^y$.</p><p>Then $9^{2y} + 4 \cdot 9^y + 3 = 0$.</p><p>Let $t = 9^y$ where $t > 0$. We get $t^2 + 4t + 3 = 0$.</p><p>Factoring: $(t+1)(t+3) = 0$, so $t = -1$ or $t = -3$.</p><p>Since $t = 9^y > 0$ for all real $y$, there are no valid solutions.</p><p>∴ The number of real values of <i>x</i> is <strong>0</strong>.</p>
Correct Answer: 0