\( 2x^2 + 6x + 9 = x^2 + 10x + 25 \)

["Solving the Equation: ( 2x^2 + 6x + 9 = x^2 + 10x + 25 )", "Solving quadratic equations is a fundamental skill in algebra, and understanding how to solve equations like ( 2x^2 + 6x + 9 = x^2 + 10x + 25 ) opens the door to many advanced math topics. In this comprehensive guide, we’ll walk through step-by-step how to simplify, solve, and interpret this equation—while also exploring its real-world relevance and modern solving techniques.", "---", "### Step 1: Simplify Both Sides", "Start by moving all terms to one side to form a standard quadratic equation equal to zero. Subtract ( x^2 + 10x + 25 ) from both sides:", "[\n2x^2 + 6x + 9 - (x^2 + 10x + 25) = 0\n]", "Distribute the minus sign and combine like terms:", "[\n2x^2 - x^2 + 6x - 10x + 9 - 25 = 0\n]", "[\nx^2 - 4x - 16 = 0\n]", "---", "### Step 2: Analyze the Quadratic Equation", "We now have the simplified standard form:", "[\nx^2 - 4x - 16 = 0\n]", "This is a quadratic equation in the form ( ax^2 + bx + c = 0 ) with:", "- ( a = 1 )\n- ( b = -4 )\n- ( c = -16 )", "---", "### Step 3: Apply the Quadratic Formula", "Since factoring isn’t immediately obvious, we use the quadratic formula:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Calculate the discriminant (( \Delta )):", "[\n\Delta = b^2 - 4ac = (-4)^2 - 4(1)(-16) = 16 + 64 = 80\n]", "Now plug values into the formula:", "[\nx = \frac{-(-4) \pm \sqrt{80}}{2(1)} = \frac{4 \pm \sqrt{80}}{2}\n]", "Simplify ( \sqrt{80} ):", "[\n\sqrt{80} = \sqrt{16 \ imes 5} = 4\sqrt{5}\n]", "So,", "[\nx = \frac{4 \pm 4\sqrt{5}}{2} = 2 \pm 2\sqrt{5}\n]", "---", "### Step 4: Final Solutions", "The equation ( 2x^2 + 6x + 9 = x^2 + 10x + 25 ) has two real solutions:", "[\nx = 2 + 2\sqrt{5} \quad \ ext{and} \quad x = 2 - 2\sqrt{5}\n]", "These are the points where the two quadratics intersect, a concept useful in optimization, physics, and economics.", "---", "### Why This Equation Matters", "This equation models situations where two quadratic relationships intersect—such as comparing cost functions, projectile motion paths, or profit margins in business. Understanding its solutions helps in:", "- Graphical analysis: Finding x-intercepts of transformed quadratics\n- Applied math: Solving for break-even points in economics\n- Algebraic technology: Implementing equation solvers in software", "---", "### Practice & Tips", "- Always simplify equations before solving.\n- Check your answer by substituting back into the original equation.\n- Use the discriminant (( b^2 - 4ac )) to predict solution types (real/two real, one real, or complex).\n- Modern tools like graphing calculators or algebra apps can verify results visually.", "---", "### Summary", "Solving ( 2x^2 + 6x + 9 = x^2 + 10x + 25 ) leads to the quadratic equation ( x^2 - 4x - 16 = 0 ), whose solutions are:", "[\nx = 2 \pm 2\sqrt{5}\n]", "These values mark the intersection points of two curves—key for modeling real-world phenomena. Whether you’re a student, educator, or enthusiast, mastering this process strengthens your algebraic foundation and problem-solving toolkit.", "---", "Keywords:\nquadratic equation, solve (2x^2 + 6x + 9 = x^2 + 10x + 25), quadratic formula, intercepts, algebra equatios, real solutions, (x^2 - 4x - 16 = 0), teach algebra, graph intersection points", "---", "Related Reading:\n- How to Graph Quadratic Functions\n- Mastering the Quadratic Formula: Techniques and Tips\n- Applications of Quadratics in Science and Engineering"]