4x² + 160x - 480 = 0

["# Solving the Quadratic Equation: 4x² + 160x - 480 = 0", "Solving quadratic equations is a fundamental skill in algebra, essential for students, engineers, scientists, and anyone interested in mathematical problem-solving. One commonly encountered equation in both theoretical and applied settings is:", "4x² + 160x - 480 = 0", "This article guides you step-by-step through solving this quadratic equation, explains how to interpret its solutions, and explores the real-world applications where such equations appear. Whether you're mastering your algebra class or applying these concepts in modeling, understanding this equation can enhance your analytical tools.", "---", "## Why Solve Quadratic Equations?", "Quadratic equations, defined by the general form ax² + bx + c = 0, model phenomena involving parabolic motion, optimization problems, and various scientific calculations. Mastering their solutions strengthens algebraic fluency and opens doors to solving more advanced problems in physics, economics, engineering, and beyond.", "---", "## Step-by-Step Solution to 4x² + 160x - 480 = 0", "### Step 1: Simplify the equation (if possible)\nHere, the equation is linear in coefficients. Factor out the greatest common factor (GCF):", "[\n4x² + 160x - 480 = 0\n]", "Divide every term by 4 to simplify:", "[\nx² + 40x - 120 = 0\n]", "Now we solve the simplified quadratic equation:\nx² + 40x - 120 = 0", "---", "### Step 2: Identify coefficients for the quadratic formula", "In the standard form ax² + bx + c = 0, we have:", "- ( a = 1 )\n- ( b = 40 )\n- ( c = -120 )", "---", "### Step 3: Apply the quadratic formula", "The quadratic formula is:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Substitute the values:", "[\nx = \frac{-40 \pm \sqrt{(40)^2 - 4(1)(-120)}}{2(1)}\n]", "Calculate inside the square root (discriminant):", "[\n\sqrt{1600 + 480} = \sqrt{2080}\n]", "Simplify √2080:", "First, factor 2080:\n(2080 = 16 \ imes 130 = 16 \ imes 10 \ imes 13)\nSo,\n[\n\sqrt{2080} = \sqrt{16 \ imes 10 \ imes 13} = 4\sqrt{130}\n]", "Now plug back in:", "[\nx = \frac{-40 \pm 4\sqrt{130}}{2}\n]", "Simplify:", "[\nx = -20 \pm 2\sqrt{130}\n]", "---", "### Step 4: Final solutions", "The two real solutions are:", "[\nx = -20 + 2\sqrt{130} \quad \ ext{and} \quad x = -20 - 2\sqrt{130}\n]", "Approximate numerical values:\n- (\sqrt{130} \approx 11.40)\n- So ( x \approx -20 + 22.80 = 2.80 )\n- And ( x \approx -20 - 22.80 = -42.80 )", "---", "## How to Interpret the Solutions", "These irrational solutions indicate that the parabola representing the equation x² + 40x - 120 = 0 crosses the x-axis at (x \approx 2.80) and (x \approx -42.80). Since the coefficient of (x^2) is positive, the parabola opens upward, and the solutions represent the x-intercepts (roots) of the function.", "---", "## Real-World Applications", "This type of equation arises in practical scenarios such as:", "- Projectile motion: Calculating the time when a projectile hits the ground.\n- Economic modeling: Finding break-even points where revenue equals cost.\n- Engineering optimization: Determining dimensions that maximize area or minimize cost under quadratic constraints.", "Understanding how to solve and interpret such equations equips you to tackle complex, real-life challenges.", "---", "## Tools for Equation Solving", "Use tools like:\n- Graphing calculators to visualize the parabola and verify solutions\n- Algebraic apps for quick quadratic formula applications\n- Manual simplification to build foundational skills", "---", "## Summary", "Solving 4x² + 160x - 480 = 0 reduces to x² + 40x - 120 = 0, which yields two real irrational roots via the quadratic formula:", "[\nx = -20 \pm 2\sqrt{130}\n]", "Mastering these steps empowers you to solve similar quadratic equations and apply them confidently across disciplines. Whether for homework, exams, or real-world modeling, quadratic equations remain a cornerstone of mathematical problem-solving.", "---", "## Additional Resources", "- Khan Academy: Quadratic Equations\n- Desmos Graphing Calculator: Input and visualize 4x² + 160x - 480\n- Algebra textbooks and online tutorials for extra practice", "---", "Keywords: quadratic equation, solve 4x² + 160x - 480, quadratic formula, x = -20 ± 2√130, algebra, solving quadratics, projectile motion, mathematical modeling"]