\[ a^2 + 81 = 225 \] - Dygne

["Solving the Equation ( a^2 + 81 = 225 ): A Step-by-Step Guide", "Solving quadratic equations is a fundamental skill in algebra, and understanding how to solve equations like ( a^2 + 81 = 225 ) can make math problems much clearer. In this article, we’ll explore how to solve the equation step-by-step, interpret its solutions, and highlight why mastering this process is essential for students and lifelong learners.", "---", "### Understanding the Equation: ( a^2 + 81 = 225 )", "The equation ( a^2 + 81 = 225 ) is a simple quadratic equation. Although it doesn’t require advanced factoring or the quadratic formula, solving it involves isolating the variable and simplifying expressions.", "---", "### Step-by-Step Solution", "#### Step 1: Isolate the Term with the Variable
\nStart by subtracting 81 from both sides of the equation to isolate the term with ( a ):", "[
\na^2 + 81 - 81 = 225 - 81
\n]", "[
\na^2 = 144
\n]", "#### Step 2: Solve for ( a )
\nTo find ( a ), take the square root of both sides:", "[
\na = \pm \sqrt{144}
\n]", "[
\na = \pm 12
\n]", "This means the equation has two real solutions: ( a = 12 ) and ( a = -12 ).", "---", "### Why This Equation Matters", "This type of equation—where a squared term plus a constant equals another constant—is commonly encountered in algebra. It serves as a foundational step toward solving more complex quadratics, such as those needing the quadratic formula or completing the square.", "Practicing such problems helps develop:", "- Algebraic manipulation skills
\n- Mastery of square roots and absolute values
\n- Ability to verify solutions", "---", "### Verification: Checking Your Answer", "It’s always a good practice to plug your solutions back into the original equation.", "- For ( a = 12 ):
\n ( 12^2 + 81 = 144 + 81 = 225 ) ✅
\n- For ( a = -12 ):
\n ( (-12)^2 + 81 = 144 + 81 = 225 ) ✅", "Both solutions are correct.", "---", "### Real-World Applications", "While ( a^2 + 81 = 225 ) may appear abstract, similar equations model real-life situations. For example:", "- Geometry: Computing side lengths in right triangles using the Pythagorean theorem.
\n- Finance: Calculating break-even points where cost and revenue relate quadratically.
\n- Physics: Modeling distances or energies in motion equations.", "---", "### How to Solve Similar Equations", "The method used here—isolating the squared term and applying square roots—can be applied broadly. For equations of the form:", "[
\na^2 + k = c
\n]", "the solution steps are:", "1. Subtract ( k ): ( a^2 = c - k )
\n2. Take square roots: ( a = \pm \sqrt{c - k} )
\n3. Ensure solutions are valid if domain restrictions exist.", "---", "### Conclusion", "Solving ( a^2 + 81 = 225 ) is more than just an algebraic exercise—it’s mastering a core problem-solving technique. By isolating the squared term and carefully applying square roots, you confidently uncover both positive and negative solutions. This skill supports deeper math comprehension and opens doors to solving advanced problems across science, engineering, and finance.", "Practice tip: Regularly solve equations like this to build fluency and speed—key for exams and real-world application.", "---", "Keywords for SEO:
\na² + 81 = 225, solve quadratic equation, isolate variable, square root solutions, algebra practice, solving equations step by step, negative solution, real-world math applications, quadratic equations foundation.", "Related reading:
\nHow to Solve Linear Equations
\nQuadratic Equations Made Easy
\nAlgebra Tips for Beginners"]