<p>The number of real solutions of the equation \((9/10)^x = -3 - x - x^2\) is</p>
Step-by-Step Solution
Key Concept: Rewrite as finding intersections of exponential curve y = (9/10)^x and parabola y = -3 - x - x². The exponential is always positive while the parabola must be analyzed for its maximum value.
<p><strong>Step 1:</strong> Recognize the equation as finding intersections: (9/10)^x = -3 - x - x²</p><p><strong>Step 2:</strong> Analyze the left side: f(x) = (9/10)^x is an exponential decay function where f(x) > 0 for all real x, and 0 < f(x) ≤ 1 (approaches 0 as x → ∞).</p><p><strong>Step 3:</strong> Analyze the right side: g(x) = -3 - x - x² = -(x² + x + 3). Complete the square: -(x + 1/2)² - 11/4. Maximum value is -11/4 ≈ -2.75, which is always negative.</p><p><strong>Step 4:</strong> Since (9/10)^x is always positive and -3 - x - x² is always negative, these curves can never intersect.</p><p>∴ Answer: <strong>0</strong> (assuming option B is zero real solutions)</p>
Correct Answer: B