Ausmultiplizieren: \(12.000 + 460x + 4x^2 = 15.120\)
["Title: Solving the Quadratic Equation: Ausmultiplizieren & Lösung von (12.000 + 460x + 4x^2 = 15.120)", "---", "Introduction\nSolving quadratic equations is a fundamental skill in algebra, essential across scientific and engineering disciplines. One common challenge students face is simplifying and solving equations in standard form, such as ( 12.000 + 460x + 4x^2 = 15.120 ). This article guides you step-by-step through the process of “Ausmultiplizieren” (expanding and rearranging), solving the quadratic, and interpreting the results. Understanding and mastering this technique unlocks deeper problem-solving abilities.", "---", "### 1. Understanding the Equation\nWe begin with:\n[\n4x^2 + 460x + 12.000 = 15.120\n]\nThis is a quadratic equation in the standard form:\n[\nax^2 + bx + c = 0\n]\nFirst, we rearrange the equation by subtracting 15.120 from both sides to set it to zero:\n[\n4x^2 + 460x + 12.000 - 15.120 = 0\n]\n[\n4x^2 + 460x - 3.120 = 0\n]", "---", "### 2. Ausmultiplizieren (Rearranging Terms)\n“Ausmultiplizieren” in this context means reordering the equation to clearly identify (a), (b), and (c):\n[\n4x^2 + 460x - 3.120 = 0\n]\nHere,\n- ( a = 4 )\n- ( b = 460 )\n- ( c = -3.120 )", "---", "### 3. Solving the Quadratic Equation\nThe standard method for solving quadratics is the quadratic formula:\n[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]\nLet’s compute each term carefully.", "#### Step 3.1: Compute the Discriminant\n[\n\Delta = b^2 - 4ac = (460)^2 - 4 \cdot 4 \cdot (-3.120)\n]\n[\n\Delta = 211600 + 49.92 = 211649.92\n]", "#### Step 3.2: Take the Square Root\n[\n\sqrt{\Delta} = \sqrt{211649.92} \approx 460.0121\n]\n(This result is very close to 460, confirming our prior coefficients.)", "#### Step 3.3: Apply the Quadratic Formula\n[\nx = \frac{-460 \pm 460.0121}{2 \cdot 4} = \frac{-460 \pm 460.0121}{8}\n]", "- First solution:\n[\nx_1 = \frac{-460 + 460.0121}{8} = \frac{0.0121}{8} \approx 0.0015125\n]\n- Second solution:\n[\nx_2 = \frac{-460 - 460.0121}{8} = \frac{-920.0121}{8} \approx -115.0018\n]", "---", "### 4. Final Interpretation\nThe equation ( 4x^2 + 460x - 3.120 = 0 ) has two real solutions:\n[\nx \approx 0.001513 \quad \ ext{and} \quad x \approx -115.002\n]\nThese values can represent real-world quantities depending on context—such as measurements, scaling factors, or system parameters.", "---", "### 5. Practical Tips for Ausmultiplizieren\n- Always start by moving all terms to one side to form ( ax^2 + bx + c = 0 ).\n- Double-check signs when subtracting constants.\n- Organize coefficients clearly before applying the quadratic formula.\n- Use a calculator for square root operations but verify approximate results through estimation.", "---", "### Conclusion\nUnderstanding how to “ausmultiplizieren” an algebraic expression and solve quadratic equations equips learners with a powerful tool applicable in physics, finance, and data analysis. Mastering steps like rearranging equations and applying the quadratic formula builds confidence in tackling complex mathematical models. Whether you're a student, educator, or professional, this approach clarity and precision in equation solving.", "---", "Keywords:\nAusmultiplizieren, quadratische Gleichung, 12.000 + 460x + 4x² = 15.120, Lösung, quadratische Formel, Gleichung lösen\nMeta Description:\nLearn how to solve (4x^2 + 460x - 3.120 = 0) using Ausmultiplizieren, step-by-step. Understand coefficients, discriminant, and real solutions step-by-step. Perfect for math students and educators.", "---", "Related Articles:\n- How to Solve Quadratic Equations Using Factoring\n- Step-by-Step: Completing the Square\n- Applications of Quadratics in Real-World Problems", "---", "Embrace the power of algebra—clear expression, precise calculation, real solutions."]
