x² + x² + 2x + 1 + x² + 4x + 4 = 425
["# Solving the Equation: x² + x² + 2x + 1 + x² + 4x + 4 = 425", "Mathematics often turns everyday problems into interesting challenges, and this quadratic equation offers a perfect blend of algebraic manipulation and logical reasoning. In this article, we’ll break down the equation step-by-step, simplify it gracefully, and solve it efficiently—perfect for learners, students, and anyone looking to sharpen their algebra skills.", "---", "## Understanding the Equation", "The given equation is:", "[\nx^2 + x^2 + 2x + 1 + x^2 + 4x + 4 = 425\n]", "### Step 1: Combine Like Terms", "Start by combining all identical terms on the left-hand side.", "- Quadratic terms: (x^2 + x^2 + x^2 = 3x^2)\n- Linear terms: (2x + 4x = 6x)\n- Constant terms: (1 + 4 = 5)", "Rewriting the equation:", "[\n3x^2 + 6x + 5 = 425\n]", "---", "## Step 2: Simplify the Equation", "Subtract 425 from both sides to set the equation to zero:", "[\n3x^2 + 6x + 5 - 425 = 0\n]", "[\n3x^2 + 6x - 420 = 0\n]", "For easier solving, divide the entire equation by 3:", "[\nx^2 + 2x - 140 = 0\n]", "---", "## Step 3: Solve the Quadratic Equation", "We now solve the simplified quadratic equation:", "[\nx^2 + 2x - 140 = 0\n]", "This is a standard quadratic in the form (ax^2 + bx + c = 0), where:", "- (a = 1)\n- (b = 2)\n- (c = -140)", "### Using the Quadratic Formula", "Recall the quadratic formula:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Plug in the values:", "[\nx = \frac{-2 \pm \sqrt{(2)^2 - 4(1)(-140)}}{2(1)}\n]", "[\nx = \frac{-2 \pm \sqrt{4 + 560}}{2}\n]", "[\nx = \frac{-2 \pm \sqrt{564}}{2}\n]", "Simplify (\sqrt{564}):", "Note that (564 = 4 \ imes 141), so:", "[\n\sqrt{564} = \sqrt{4 \ imes 141} = 2\sqrt{141}\n]", "Now substitute back:", "[\nx = \frac{-2 \pm 2\sqrt{141}}{2} = -1 \pm \sqrt{141}\n]", "---", "## Step 4: Find Exact Solutions", "Thus, the two real solutions are:", "[\nx = -1 + \sqrt{141} \quad \ ext{and} \quad x = -1 - \sqrt{141}\n]", "### Approximate Values (for reference):", "- (\sqrt{141} \approx 11.87)\n- (x \approx -1 + 11.87 = 10.87)\n- (x \approx -1 - 11.87 = -12.87)", "These are the decimal approximations, but the exact form remains preferred in algebraic solutions.", "---", "## Why This Equation Matters", "Though seemingly abstract, this equation models real-world scenarios—such as optimizing profit margins, analyzing motion paths, or designing geometric shapes—where quadratic relationships arise naturally. Learning to solve such equations builds a strong foundation in algebra and enhances problem-solving precision.", "---", "## Final Thoughts", "Solving (x^2 + x^2 + 2x + 1 + x^2 + 4x + 4 = 425) involves careful simplification, applying the quadratic formula correctly, and interpreting both exact and approximate solutions. With patience and method, even complex quadratics become manageable.", "Keywords:\nx² quadratic equation, solve 3x² + 6x + 5 = 425, quadratic formula application, solve x² + 2x - 140 = 0, simplifying algebraic expressions, step-by-step equation solving, algebra practice problem, exact and approximate solutions", "---", "Try solving similar quadratics to master this skill—your confidence in math grows with every equation!"]
