The square of a number exceeds twice the number by 15. What is the number?
["# The Square of a Number Exceeds Twice the Number by 15 — What Is the Number?", "Have you ever wondered what number, when squared, exceeds twice the number by exactly 15? This is a classic algebraic puzzle that blends logic and equation-solving in a simple yet elegant way. Let’s break it down step-by-step to find the number.", "## Understanding the Problem", "We are told:\nThe square of a number exceeds twice the number by 15.", "Let the unknown number be represented by ( x ).\n- The square of the number is ( x^2 ).\n- Twice the number is ( 2x ).\n- The statement translates mathematically to:\n[\nx^2 = 2x + 15\n]", "This equation sets the square equal to twice the number plus 15, meaning the difference between ( x^2 ) and ( 2x ) is 15.", "---", "## Solving the Equation", "We begin by rearranging the equation into standard quadratic form:\n[\nx^2 - 2x - 15 = 0\n]", "Now, we solve this quadratic equation using factoring, completing the square, or the quadratic formula. Here, factoring works nicely.", "### Factoring", "We look for two numbers that multiply to (-15) and add to (-2). These numbers are (-5) and (3).\nThus:\n[\nx^2 - 2x - 15 = (x - 5)(x + 3) = 0\n]", "Setting each factor equal to zero gives:\n[\nx - 5 = 0 \quad \Rightarrow \quad x = 5\n]\n[\nx + 3 = 0 \quad \Rightarrow \quad x = -3\n]", "---", "## Verifying the Solutions", "Both ( x = 5 ) and ( x = -3 ) satisfy the original condition. Let’s check both:", "- For ( x = 5 ):\n ( x^2 = 25 ),\n ( 2x + 15 = 2(5) + 15 = 10 + 15 = 25 ) ✅", "- For ( x = -3 ):\n ( x^2 = 9 ),\n ( 2x + 15 = 2(-3) + 15 = -6 + 15 = 9 ) ✅", "Both numbers produce a perfect balance — their squares equal twice themselves plus 15.", "---", "## What’s the Correct Answer?", "Since the problem asks "What is the number?" and does not restrict to positive values, both 5 and -3 are valid solutions. However, depending on context:", "- If only one answer is expected, the positive solution ( \mathbf{5} ) is most commonly sought in elementary problems.\n- In broader mathematical contexts, both 5 and -3 are acceptable.", "Still, in classic word problems like this, positive integers are typically emphasized.", "---", "## Final Answer", "The number(s) whose square exceeds twice the number by 15 are:\n[\n\boxed{5} \quad \ ext{and} \quad \boxed{-3}\n]", "For simplicity and context, the number is 5 — a clean, positive solution to this intriguing equation.", "---", "## Want to Explore More?", "Try solving similar equations:\n- A number plus 7 squared equals twice the number plus 21?\n- Or, find a number such that its square is five times itself plus 14?", "These will sharpen your algebraic intuition and problem-solving speed!", "---", "### SEO Optimization Notes\n- Target Keywords: square of a number, quadratic equation, solve algebra, number that square exceeds twice itself by 15\n- Clear structure: Step-by-step solution improves readability and user engagement\n- Engaging question: Opens curiosity and invites active problem-solving\n- Multiple ends: Provides both solutions with context for deeper understanding", "Optimized for both search engines and clear learning outcomes."]
