The area is \(x(x + 50) = 6000\).
["# Solving the Quadratic Equation: ( x(x + 50) = 6000 )", "Finding solutions to quadratic equations is a fundamental step in algebra, helping students, professionals, and math enthusiasts alike. One commonly encountered problem is the equation ( x(x + 50) = 6000 ). This article breaks down how to solve this equation step-by-step, explains its real-world context, and provides insights into why these types of problems matter.", "## Understanding the Equation: ( x(x + 50) = 6000 )", "The given equation is:", "[ x(x + 50) = 6000 ]", "First, expand the left-hand side:", "[ x^2 + 50x = 6000 ]", "Next, rearrange the equation into standard quadratic form:", "[ x^2 + 50x - 6000 = 0 ]", "This is a quadratic equation of the form ( ax^2 + bx + c = 0 ), where ( a = 1 ), ( b = 50 ), and ( c = -6000 ).", "## Solving the Quadratic Equation", "There are several methods to solve ( x^2 + 50x - 6000 = 0 ): factoring, completing the square, or using the quadratic formula.", "### Method 1: Factoring", "We attempt to factor the quadratic. We need two numbers that multiply to ( -6000 ) and add to ( 50 ).", "After testing factor pairs of ( -6000 ), we find:", "- ( 100 \ imes (-60) = -6000 ), and ( 100 + (-60) = 40 ) → not correct\n- ( 120 \ imes (-50) = -6000 ), and ( 120 + (-50) = 70 ) → no\n- ( 150 \ imes (-40) = -6000 ), and ( 150 + (-40) = 110 ) → no\n- ( 200 \ imes (-30) = -6000 ), and ( 200 + (-30) = 170 ) → no \nEventually, trying ( (x + 100)(x - 60) = x^2 + 40x - 6000 ), not exact.", "Instead, fully apply the quadratic formula for reliability:", "### Method 2: Quadratic Formula", "The quadratic formula is:", "[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ]", "Plug in ( a = 1 ), ( b = 50 ), ( c = -6000 ):", "[ x = \frac{-50 \pm \sqrt{50^2 - 4(1)(-6000)}}{2(1)} ]\n[ x = \frac{-50 \pm \sqrt{2500 + 24000}}{2} ]\n[ x = \frac{-50 \pm \sqrt{26500}}{2} ]\n[ x = \frac{-50 \pm \sqrt{26500}}{2} ]", "Simplify ( \sqrt{26500} ):", "[ \sqrt{26500} = \sqrt{100 \ imes 265} = 10\sqrt{265} ]", "So,", "[ x = \frac{-50 \pm 10\sqrt{265}}{2} ]\n[ x = -25 \pm 5\sqrt{265} ]", "These are the two real solutions.", "---", "## Approximate Numerical Solutions", "To get a clearer numerical sense:", "[ \sqrt{265} \approx 16.2788 ]\n[ 5\sqrt{265} \approx 81.394 ]", "Thus:\n[ x \approx -25 + 81.394 = 56.394 ]\n[ x \approx -25 - 81.394 = -106.394 ]", "Since ( x ) typically represents a physical quantity like length or time, the meaningful solution is ( x \approx 56.39 ) (rounded).", "---", "## Real-World Applications and Why This Equation Matters", "Equations like ( x(x + 50) = 6000 ) often model real-life scenarios. For example:", "- Engineering: Calculating dimensions of materials where area or volume relates linearly and quadratically.\n- Finance: Estimating investment growth over time with compounding effects.\n- Construction: Determining the length of materials needed when constraints involve total area or combined measurements.", "这种二次方程不仅锻炼代数技能，更培养逻辑思维与问题解决能力。", "---", "## Summary", "To solve ( x(x + 50) = 6000 ):", "1. Expand to ( x^2 + 50x - 6000 = 0 ).\n2. Apply the quadratic formula: ( x = \frac{-50 \pm \sqrt{26500}}{2} ).\n3. Simplify to approximate ( x \approx 56.39 ) (positive solution).", "Understanding and solving such equations empowers you to tackle complex real-world problems with confidence.", "---", "Keywords: quadratic equation, solve (x(x + 50) = 6000), algebraic methods, quadratic formula, real-world applications, algebra learning."]
