x^2 + (x + 2)^2 = 16

["# Solve the Equation: x² + (x + 2)² = 16", "Understanding how to solve quadratic equations is essential for mastering algebra. One common challenge often encountered is solving expressions like x² + (x + 2)² = 16. Whether you're a student, teacher, or math enthusiast, learning how to simplify and solve this equation step-by-step can boost your problem-solving skills and confidence in handling quadratic expressions.", "### What is the Equation?\nThe equation x² + (x + 2)² = 16 combines two squared binomials and includes a constant term. It represents a classic quadratic identity that appears in many algebra and calculus problems, and it's an excellent example of applying the expansion and simplification techniques for solving equations.", "---", "## Step 1: Expand the Squared Term\nStart by expanding (x + 2)² using the binomial square formula:", "[\n(x + 2)^2 = x^2 + 4x + 4\n]", "Now substitute this into the original equation:", "[\nx^2 + (x^2 + 4x + 4) = 16\n]", "---", "## Step 2: Simplify the Expression\nCombine like terms:", "[\nx^2 + x^2 + 4x + 4 = 16\n]\n[\n2x^2 + 4x + 4 = 16\n]", "Subtract 16 from both sides to set the equation to zero:", "[\n2x^2 + 4x + 4 - 16 = 0\n]\n[\n2x^2 + 4x - 12 = 0\n]", "---", "## Step 3: Simplify Further\nDivide the entire equation by 2 to simplify:", "[\nx^2 + 2x - 6 = 0\n]", "Now you’re dealing with a standard quadratic equation:\nx² + 2x – 6 = 0", "---", "## Step 4: Solve Using the Quadratic Formula\nFor equations of the form ax² + bx + c = 0, the quadratic formula gives:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Here, a = 1, b = 2, and c = -6. Plug in these values:", "[\nx = \frac{-2 \pm \sqrt{(2)^2 - 4(1)(-6)}}{2(1)} = \frac{-2 \pm \sqrt{4 + 24}}{2} = \frac{-2 \pm \sqrt{28}}{2}\n]", "Since √28 = √(4×7) = 2√7, this simplifies to:", "[\nx = \frac{-2 \pm 2\sqrt{7}}{2} = -1 \pm \sqrt{7}\n]", "---", "## Step 5: Final Answer\nThus, the solutions to the equation x² + (x + 2)² = 16 are:", "[\nx = -1 + \sqrt{7} \quad \ ext{and} \quad x = -1 - \sqrt{7}\n]", "These exact roots reflect the two points where the parabola defined by the left-hand side intersects the horizontal line y = 16.", "---", "## Why This Equation Matters\nSolving equations like x² + (x + 2)² = 16 helps strengthen skills in algebraic expansion, simplification, and quadratic formula application. These problems frequently appear in physics, engineering, and optimization contexts where we model distances, areas, and curves.", "---", "## Practice & Real-World Usage\nTry solving:\n- x² + (x – 1)² = 10\n- (x – 3)² + x² = 25", "Such problems also emerge in graph transformations and distance calculations, making them practically useful beyond classroom exercises.", "---", "### Summary\nSolving x² + (x + 2)² = 16 involves expanding binomials, combining like terms, factoring, and applying the quadratic formula. The final solutions are:", "[\nx = -1 + \sqrt{7} \quad \ ext{and} \quad x = -1 - \sqrt{7}\n]", "Mastering this process equips you to tackle more complex quadratic problems with ease and confidence. Keep practicing — algebraic fluency comes with repetition!", "---", "### Key Themes / Keywords for SEO\n- Solve x² + (x + 2)² = 16\n- Quadratic equation solved step-by-step\n- Algebra practice problems\n- Learn quadratic formulas and simplification\n- Mathematics tutoring tips\n- Quadratic functions and solutions", "---", "If you found this guide helpful, share it with classmates or bookmark it for future review—algebra just got easier!"]