["# Solve ( x^2 + 64 = 100 ): Step-by-Step Guide and Key Insights", "Solving quadratic equations is a fundamental skill in algebra, and one of the most common beginner problems is solving ( x^2 + 64 = 100 ). This equation appears simple but offers an excellent opportunity to strengthen algebraic skills and understand core math concepts. In this SEO-optimized guide, we’ll break down the solution, explain how to solve similar problems, and explore real-world applications for better retention.", "## What is the Equation ( x^2 + 64 = 100 )?", "The equation ( x^2 + 64 = 100 ) is a quadratic equation in standard form, though it can be rewritten to standard quadratic structure as ( x^2 - 36 = 0 ). It models a simple relationship where squaring a number ( x ) and adding 64 yields 100. Solving it involves isolating ( x^2 ), then applying square roots to find all real solutions.", "---", "## Step-by-Step Solution", "### Step 1: Isolate the ( x^2 ) Term
\nStart by subtracting 64 from both sides of the equation:
\n[
\nx^2 + 64 - 64 = 100 - 64
\n]
\nSimplifying:
\n[
\nx^2 = 36
\n]", "### Step 2: Solve Using Square Roots
\nTo eliminate the square, take the square root of both sides. Remember, both the positive and negative roots are valid:
\n[
\nx = \pm \sqrt{36}
\n]
\nThus:
\n[
\nx = 6 \quad \ ext{or} \quad x = -6
\n]", "---", "## Key Concepts Behind the Solution", "### Why ( x = \pm \sqrt{36} )?
\nBecause squaring a number means multiplying it by itself, both ( 6 ) and ( -6 ) produce the same positive result:
\n[
\n6^2 = 36, \quad (-6)^2 = 36
\n]
\nThis key insight applies broadly in solving quadratic equations.", "### Verifying Solutions
\nAlways substitute values back into the original equation to confirm correctness.
\n- For ( x = 6 ): ( 6^2 + 64 = 36 + 64 = 100 ) ✅
\n- For ( x = -6 ): ( (-6)^2 + 64 = 36 + 64 = 100 ) ✅", "---", "## Real-World Applications of ( x^2 + 64 = 100 )", "Equation-solving skills extend beyond the classroom. For example, in physics, ( x^2 ) might represent a squared velocity or distance, and such equations model motion, energy, or design. Understanding these basics enables tackling more complex problems in engineering, economics, or data science.", "---", "## Common Errors to Avoid", "- Forgetting to subtract 64, leaving ( x^2 = 100 ), which yields ( x = \pm 10 ) — a classic mix-up.
\n- Skipping verification, which risks accepting incorrect solutions.", "---", "## Final Answer", "The solutions to ( x^2 + 64 = 100 ) are:
\n[
\nx = 6 \quad \ ext{and} \quad x = -6
\n]
\nMastering this problem builds a foundation for solving more advanced quadratics and symbolizes the power of systematic algebraic thinking.", "---", "## Optimized Keywords for SEO
\n- Solve ( x^2 + 64 = 100 ) step-by-step
\n- Quadratic equation solution
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\n- Algebraic problem-solving guide", "By mastering this equation, you unlock broader math proficiency and enhance your ability to solve practical, real-world challenges!", "---
\nKeywords: x² + 64 = 100, solve quadratic equations, step-by-step method, real root solutions, verify quadratic solutions"]
