["# Solving ( a^2 + 64 = 100 ): A Comprehensive Guide to Quadratic Equations", "Understanding how to solve equations like ( a^2 + 64 = 100 ) is fundamental in algebra and forms the basis for mastering quadratic expressions. This article explores the step-by-step process of solving the equation ( a^2 + 64 = 100 ), explains key algebraic concepts, and presents practical applications of this simple yet essential equation.", "## Understanding the Equation ( a^2 + 64 = 100 )", "The equation ( a^2 + 64 = 100 ) is a quadratic equation in its simplest form. Although it includes a squared term (( a^2 )), it can be treated similarly to a linear equation since it’s set to 100, a constant. The goal is to isolate ( a ) and find all real or complex solutions.", "### Step-by-Step Solution", "### Step 1: Subtract 64 from Both Sides
\nTo eliminate the constant term on the left, subtract 64 from both sides:", "[
\na^2 + 64 - 64 = 100 - 64
\n]", "[
\na^2 = 36
\n]", "### Step 2: Take the Square Root of Both Sides
\nTo solve for ( a ), take the square root:", "[
\na = \pm \sqrt{36}
\n]", "[
\na = \pm 6
\n]", "### Results
\nThe equation ( a^2 + 64 = 100 ) has two real solutions:
\n- ( a = 6 )
\n- ( a = -6 )", "---", "## Why This Equation Matters", "Even though ( a^2 + 64 = 100 ) appears straightforward, it illustrates core algebraic principles:", "- Isolation of Variable: The process demonstrates how isolating variables using inverse operations simplifies equations.
\n- Quadratic Nature: The squared term introduces quadratic behavior—solving quadratics is critical in physics, engineering, computer science, and economics.
\n- Equality Preservation: Every step maintains equation balance, a fundamental rule in algebra.", "---", "## How to Solve Similar Equations", "### General Form
\nAny equation of the form:", "[
\na^2 + C = D
\n]", "can be solved by subtracting ( C ) and taking square roots:", "[
\na^2 = D - C \quad \Rightarrow \quad a = \pm \sqrt{D - C}
\n]", "- Ensure ( D - C \geq 0 ) for real solutions.
\n- For complex numbers, use ( a = \pm i\sqrt ) if ( D - C < 0 ).", "---", "## Real-World Applications", "### Physics
\nIn motion equations, quadratic forms like ( s = ut + \frac{1}{2}at^2 ) often lead to equations resembling ( a^2 ). Solving for time or velocity frequently involves handling squared variables.", "### Finance
\nQuadratic equations model profit maximization, cost functions, and investment returns, where squared terms appear naturally.", "### Geometry
\nCalculating dimensions when area or diagonal measurements are known often results in equations involving squared unknowns.", "---", "## Final Thoughts", "Solving ( a^2 + 64 = 100 ) may seem elementary, but it forms the foundation of quadratic algebra. Mastering such equations strengthens your ability to tackle complex mathematical models across science, engineering, and finance. Use systematic steps—subtracting constants and taking square roots—to confidently solve any quadratic equation.", "### Practice Problem", "Try solving:
\n[
\na^2 + 25 = 50
\n]", "Solution:
\nSubtract 25 from both sides:
\n[
\na^2 = 25
\n]
\nTake square roots:
\n[
\na = \pm 5
\n]
\n✓ Solutions: ( a = 5 ), ( a = -5 )", "---", "## Key Takeaways", "- Subtract constants to isolate squared terms.
\n- Square roots yield both positive and negative solutions.
\n- This equation models real-world scenarios involving quadratic relationships.", "Start practicing today—solving quadratic equations unlocks deeper mathematical fluency!"]
