Solving \(x^2 + 50x - 6000 = 0\) using the quadratic formula: \(x = \frac{-50 \pm \sqrt{50^2 + 4 \times 6000}}{2}\).
["How to Solve the Quadratic Equation (x^2 + 50x - 6000 = 0) Using the Quadratic Formula", "Solving quadratic equations is a fundamental skill in algebra, widely applicable in physics, engineering, economics, and many other fields. One of the most reliable methods for finding the roots of a quadratic equation is the quadratic formula. In this article, we’ll walk step-by-step through solving the equation:", "[\nx^2 + 50x - 6000 = 0\n]", "using the quadratic formula, showing how careful application of the formula leads to accurate solutions.", "---", "### Understanding the Quadratic Formula", "For any quadratic equation in the standard form:", "[\nax^2 + bx + c = 0\n]", "the solutions are given by:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "In our equation,\n- (a = 1)\n- (b = 50)\n- (c = -6000)", "Substituting these into the formula yields:", "[\nx = \frac{-50 \pm \sqrt{50^2 - 4(1)(-6000)}}{2(1)} = \frac{-50 \pm \sqrt{50^2 + 4 \ imes 6000}}{2}\n]", "This matches the exact expression used—critical for accurate results.", "---", "### Step 1: Calculate the Discriminant", "The first part of the formula is the discriminant:", "[\n\Delta = b^2 - 4ac = 50^2 - 4(1)(-6000) = 2500 + 24000 = 26500\n]", "A positive discriminant confirms two distinct real solutions.", "---", "### Step 2: Compute the Square Root of the Discriminant", "Now compute:", "[\n\sqrt{26500}\n]", "Note that (26500 = 100 \ imes 265), so:", "[\n\sqrt{26500} = \sqrt{100 \ imes 265} = 10\sqrt{265} \approx 162.63\n]", "For higher precision, use a calculator:", "[\n\sqrt{26500} \approx 162. Strat, but keep exact form for now:\n[\n\sqrt{26500} = 10\sqrt{265}\n]", "---", "### Step 3: Plug Back into the Formula", "Now substitute back:", "[\nx = \frac{-50 \pm \sqrt{26500}}{2} = \frac{-50 \pm 10\sqrt{265}}{2}\n]", "Then simplify:", "[\nx = -25 \pm 5\sqrt{265}\n]", "---", "### Final Solutions", "Thus, the two solutions are:", "[\nx_1 = -25 + 5\sqrt{265} \approx -25 + 162.63 = 137.63\n]\n[\nx_2 = -25 - 5\sqrt{265} \approx -25 - 162.63 = -187.63\n]", "---", "### Why This Method Works", "Using the quadratic formula ensures you avoid arithmetic errors common when factoring, especially with non-integer coefficients. The step-by-step breakdown—computing discriminant, square root, and final division—makes the solution transparent and verifiable.", "---", "### Summary", "Solving (x^2 + 50x - 6000 = 0) using the quadratic formula gives:", "[\n\boxed{x = -25 \pm 5\sqrt{265}}\n]", "These exact solutions can be approximated numerically if needed, but the precise form ensures accuracy in further calculations.", "---", "### SEO Keywords", "- Solve quadratic equation (x^2 + 50x - 6000 = 0)\n- Quadratic formula step-by-step\n- How to find real roots with discriminant\n- Solve (x^2 + 50x - 6000 = 0) exactly and numerically\n- Algebraic methods for quadratic equations", "---", "By mastering this approach, you empower yourself to tackle any quadratic equation with confidence and precision. Use the formula correctly, compute the discriminant carefully, and embrace the power of exact and numerical solutions alike."]
