Solving: 12,000 + 460x + 4x² = 15,000

["Solving the Quadratic Equation: 12,000 + 460x + 4x² = 15,000 – A Step-by-Step Guide", "When faced with a quadratic equation like 12,000 + 460x + 4x² = 15,000, solving it correctly requires a clear understanding of algebraic techniques. Whether you're studying math, preparing for an exam, or solving a real-world problem, knowing how to resolve such equations efficiently is invaluable. This article guides you through the step-by-step solution of this quadratic equation, emphasizing clarity and proper methods.", "---", "### Understanding the Equation", "We begin by rewriting the equation in standard quadratic form:\n4x² + 460x + 12,000 = 15,000", "To simplify, subtract 15,000 from both sides:\n4x² + 460x + 12,000 - 15,000 = 0\n4x² + 460x - 3,000 = 0", "Now we have a standard quadratic in the form:\nax² + bx + c = 0\nwhere\n- ( a = 4 )\n- ( b = 460 )\n- ( c = -3,000 )", "---", "### Step 1: Simplify the Equation", "Since the coefficients are large, divide every term by the greatest common factor (GCF) to simplify. Here, the GCF of 4, 460, and 3,000 is 2:", "(4x² ÷ 2) + (460x ÷ 2) + (-3,000 ÷ 2) = 0\n2x² + 230x - 1,500 = 0", "---", "### Step 2: Apply the Quadratic Formula", "For any quadratic equation ( ax² + bx + c = 0 ), the solutions are given by:\n[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Plug in ( a = 2 ), ( b = 230 ), ( c = -1,500 ):\n[\nx = \frac{-230 \pm \sqrt{(230)^2 - 4(2)(-1,500)}}{2(2)}\n]", "---", "### Step 3: Calculate the Discriminant", "First calculate the discriminant ( D ):\n[\nD = b^2 - 4ac = 230^2 - 4(2)(-1,500)\n]\n[\nD = 52,900 + 12,000 = 64,900\n]", "Since the discriminant is positive, there are two distinct real solutions.", "---", "### Step 4: Take the Square Root of the Discriminant", "[\n\sqrt{D} = \sqrt{64,900} = 254.75 , (\ ext{approximately})\n]", "Note: Because ( 254.75^2 = 64,900.5625 ), for precision, use calculator-level accuracy if needed. Here, approximation suffices for final answers.", "---", "### Step 5: Solve for ( x )", "Now substitute back into the quadratic formula:\n[\nx = \frac{-230 \pm 254.75}{4}\n]", "Calculate both possible solutions:", "First solution:\n[\nx = \frac{-230 + 254.75}{4} = \frac{24.75}{4} = 6.1875\n]", "Second solution:\n[\nx = \frac{-230 - 254.75}{4} = \frac{-484.75}{4} = -121.1875\n]", "---", "### Step 6: Final Answer", "The solutions to the equation 12,000 + 460x + 4x² = 15,000 are:\n[\n\boxed{x \approx 6.1875 \quad} \ ext{and} \quad \boxed{x \approx -121.1875}\n]", "---", "### Why This Matters", "Quadratic equations model various real-world scenarios—projectile motion, profit maximization, and geometric problems. Being able to solve such equations accurately helps students, engineers, and data analysts tackle complex problems with confidence.", "---", "### Practice Tip:", "Always verify by plugging your solutions back into the original equation. This ensures correctness and strengthens your understanding of quadratic relationships.", "---", "In summary, transforming the original equation into standard form, simplifying where possible, applying the quadratic formula, and checking your work are key steps in solving quadratic formulas cleanly and accurately. With practice, solving equations like this becomes straightforward and efficient."]