Let the number be x. Equation: x² = 2x + 15 → x² - 2x - 15 = 0.
["# Solving the Equation: x² = 2x + 15 — A Step-by-Step Guide (with x² - 2x - 15 = 0)", "Understanding how to solve quadratic equations is one of the foundational skills in algebra. One common problem students encounter involves equations in the form x² = 2x + 15, which students are often guided to rewrite as x² - 2x - 15 = 0 for easier solution. In this article, we’ll explore how to solve this equation step-by-step and highlight its importance in mathematics.", "---", "## Understanding the Equation", "The original equation is:\nx² = 2x + 15", "This expresses a quadratic relationship. To solve for x, we want to bring all terms to one side, forming a standard quadratic equation set to zero. Rearranging gives:\nx² - 2x - 15 = 0", "This follows the standard quadratic form:\nax² + bx + c = 0,\nwhere:\n- a = 1\n- b = -2\n- c = -15", "---", "## Step-by-Step Solution", "### Step 1: Identify coefficients\nFrom the equation x² - 2x - 15 = 0, identify:\n- a = 1\n- b = -2\n- c = -15", "### Step 2: Apply the quadratic formula\nThe most reliable method for solving quadratic equations is the quadratic formula:\n$$\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n$$", "Substitute the values:\n- b = -2 → $-b = 2$\n- b² = (-2)² = 4\n- 4ac = 4 × 1 × (–15) = –60", "Now calculate the discriminant:\n$$\n\Delta = b^2 - 4ac = 4 - (4 × 1 × –15) = 4 + 60 = 64\n$$", "Since Δ > 0, there are two distinct real solutions.", "### Step 3: Compute the square root of the discriminant\n$$\n\sqrt{\Delta} = \sqrt{64} = 8\n$$", "### Step 4: Plug values into the quadratic formula\n$$\nx = \frac{2 \pm 8}{2 \ imes 1} = \frac{2 \pm 8}{2}\n$$", "Now calculate the two solutions:\n- First solution:\n$$\nx = \frac{2 + 8}{2} = \frac{10}{2} = 5\n$$\n- Second solution:\n$$\nx = \frac{2 - 8}{2} = \frac{-6}{2} = -3\n$$", "---", "## Final Solutions", "The equation x² = 2x + 15 has two real solutions:\nx = 5 and x = –3", "These answers mean:\n- When the number is 5, it satisfies the original equation.\n- When the number is –3, it also satisfies the original equation.", "---", "## Why This Matters", "Quadratic equations like x² - 2x - 15 = 0 appear in physics, economics, engineering, and computer science. Solving them helps in modeling real-world phenomena such as projectile motion, profit optimization, or geometric calculations.", "---", "## Summary", "By rewriting x² = 2x + 15 as x² - 2x - 15 = 0, we transformed a simple equation into a solvable quadratic. Using the quadratic formula correctly leads to solutions x = 5 and x = –3. This process reinforces essential algebra skills and builds confidence in handling more complex equations.", "---", "## Want to Master Quadratic Equations?", "Practice solving similar equations regularly. Use factoring, completing the square, or the quadratic formula depending on the context. Tools like graphing calculators and online algebra solvers can help visualize roots and deepen your understanding.", "---", "Keywords:\nquadratic equation, solve x² = 2x + 15, x² - 2x - 15 = 0, quadratic formula, step-by-step solving, algebra skills, real solutions to quadratics"]
