Question: Determine the number of real solutions to the equation $ \sin^2(2x) + 3\sin(2x) + 2 = 0 $.
["SEO-Optimized Article: Determining the Number of Real Solutions to the Equation\n$ \sin^2(2x) + 3\sin(2x) + 2 = 0 $", "---", "### Introduction", "Understanding how many real solutions an equation has is essential in trigonometric analysis and applied mathematics. One commonly studied equation is:", "$$\n\sin^2(2x) + 3\sin(2x) + 2 = 0\n$$", "This equation combines algebraic structure with trigonometric functions, making it an ideal candidate for exploring real solutions. In this article, we will analyze the equation step by step, transform it into a more manageable form, solve it algebraically, and determine exactly how many real solutions exist.", "---", "### Step 1: Substitution to Simplify the Equation", "Let’s make a substitution to simplify the equation. Set:", "$$\nu = \sin(2x)\n$$", "Then the equation becomes:\n$$\nu^2 + 3u + 2 = 0\n$$", "This is a quadratic equation in $ u $, which is much easier to solve.", "---", "### Step 2: Solve the Quadratic Equation", "We solve:\n$$\nu^2 + 3u + 2 = 0\n$$", "Factoring:\n$$\n(u + 1)(u + 2) = 0\n$$", "Thus, the solutions are:\n$$\nu = -1 \quad \ ext{or} \quad u = -2\n$$", "---", "### Step 3: Analyze the Trigonometric Validity", "Recall that $ u = \sin(2x) $, and the sine function is bounded:\n$$\n-1 \leq \sin(2x) \leq 1\n$$", "Now evaluate the roots:\n- $ u = -1 $ → valid (within the range)\n- $ u = -2 $ → invalid, because $ -2 < -1 $, outside the range of sine", "Thus, only $ \sin(2x) = -1 $ yields real solutions.", "---", "### Step 4: Solve $ \sin(2x) = -1 $", "We now solve:\n$$\n\sin(2x) = -1\n$$", "The general solution for $ \sin \ heta = -1 $ is:\n$$\n\ heta = \frac{3\pi}{2} + 2\pi n \quad \ ext{for integer } n\n$$", "Substitute $ \ heta = 2x $:\n$$\n2x = \frac{3\pi}{2} + 2\pi n\n$$", "Solve for $ x $:\n$$\nx = \frac{3\pi}{4} + \pi n \quad \ ext{for } n \in \mathbb{Z}\n$$", "---", "### Step 5: Count the Number of Real Solutions", "Since $ n $ is any integer (positive, negative, or zero), there are infinitely many solutions of the form:\n$$\nx = \frac{3\pi}{4} + \pi n\n$$", "However, the key insight is that sine is periodic with period $ 2\pi $, so the pattern repeats every $ \pi $ units due to the factor $ 2x $. Still, each value of $ n $ gives a distinct real solution.", "Thus, the number of real solutions is infinite, but countably infinite.", "---", "### SEO-Optimized Keywords & Phrases\n- “number of real solutions to $ \sin^2(2x) + 3\sin(2x) + 2 = 0 $”\n- “Solve $ \sin^2(2x) + 3\sin(2x) + 2 = 0 $”\n- “Trig equation: how many real solutions”\n- “analysis of $ \sin(2x) = -1 $”\n- “periodicity and solution counting in trigonometric equations”", "---", "### Practical Implications", "This equation exemplifies how trigonometric identities reduce complex expressions, enabling precise determination of solution sets. In applied fields such as signal processing, vibration analysis, or oscillatory systems, recognizing the finite nature of solutions helps in modeling and prediction.", "---", "### Conclusion", "We have determined that the equation\n$$\n\sin^2(2x) + 3\sin(2x) + 2 = 0\n$$\nholds only when $ \sin(2x) = -1 $, leading to infinitely many real solutions given by:\n$$\nx = \frac{3\pi}{4} + \pi n, \quad n \in \mathbb{Z}\n$$", "Thus, the number of real solutions is infinite, with a repeating pattern every $ \pi $ units.", "---", "Meta Description:\nDiscover how many real solutions satisfy $ \sin^2(2x) + 3\sin(2x) + 2 = 0 $. We analyze the equation step-by-step, solve for $ \sin(2x) $, verify domain constraints, and conclude that the equation has infinitely many real solutions spaced by $ \pi $.", "Keywords:\n$ \sin^2(2x) + 3\sin(2x) + 2 = 0 $, real solutions, sine equation, trigonometric solutions, periodic functions, infinite countable solutions, mathematical analysis.", "---", "Checklist for SEO:\n- Title optimized with primary keyword\n- Clear structure with semantic headings\n- Relevant keywords naturally embedded\n- Practical summary and context added for user engagement\n- Technical accuracy confirmed", "Enhance your understanding and solve similar trigonometric equations with confidence!"]
