\[4x^2 + 100x + 600 = 792\]

["# Solving the Quadratic Equation: 4x² + 100x + 600 = 792", "Finding solutions to quadratic equations is a fundamental skill in algebra, widely applied in science, engineering, economics, and data modeling. In this detailed guide, we’ll solve the equation [4x^2 + 100x + 600 = 792], break down the process step-by-step, and explore practical techniques to handle quadratic expressions effectively.", "---", "## Step-by-Step Solution", "### Step 1: Rearranging the Equation", "Start by moving all terms to one side to form a standard quadratic equation equal to zero:", "[\n4x^2 + 100x + 600 - 792 = 0\n]", "Simplify:", "[\n4x^2 + 100x - 192 = 0\n]", "---", "### Step 2: Simplifying the Equation", "To make calculations easier, simplify the equation by dividing every term by the greatest common divisor (GCD) of the coefficients. Here, GCD of 4, 100, and 192 is 4:", "[\n\frac{4x^2}{4} + \frac{100x}{4} - \frac{192}{4} = 0\n]", "Which simplifies to:", "[\nx^2 + 25x - 48 = 0\n]", "---", "### Step 3: Using the Quadratic Formula", "The standard form of a quadratic equation is (ax^2 + bx + c = 0). For our simplified equation:", "- (a = 1)\n- (b = 25)\n- (c = -48)", "Apply the quadratic formula:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Calculate the discriminant ((D)):", "[\nD = b^2 - 4ac = 25^2 - 4(1)(-48) = 625 + 192 = 817\n]", "Now plug values into the formula:", "[\nx = \frac{-25 \pm \sqrt{817}}{2}\n]", "Since (\sqrt{817}) is not a perfect square, the solutions are irrational:", "[\nx = \frac{-25 + \sqrt{817}}{2} \quad \ ext{and} \quad x = \frac{-25 - \sqrt{817}}{2}\n]", "---", "### Step 4: Approximate Solutions", "For practical use, approximate (\sqrt{817} \approx 28.6):", "[\nx \approx \frac{-25 + 28.6}{2} = \frac{3.6}{2} = 1.8\n]", "[\nx \approx \frac{-25 - 28.6}{2} = \frac{-53.6}{2} = -26.8\n]", "So, the approximate solutions are:", "[\nx \approx 1.8 \quad \ ext{and} \quad x \approx -26.8\n]", "---", "## Why Solving Quadratics Matters", "Quadratic equations model real-world phenomena such as projectile motion, area optimization problems, and financial growth projections. Mastering techniques like rearranging forms and applying the quadratic formula equips learners with powerful tools for problem-solving in STEM fields and everyday applications.", "---", "## Tips for Easier Solving", "- Always rearrange equations to standard form (ax^2 + bx + c = 0).\n- Simplify coefficients where possible by dividing by GCD to reduce complexity.\n- For equations without integer roots, embrace irrational solutions or use numerical methods.\n- Use the quadratic formula confidently — it’s reliable for all discriminant signs.\n- Verify solutions by substituting back into the original equation.", "---", "## Conclusion", "Solving (4x^2 + 100x + 600 = 792) leads to the quadratic equation (x^2 + 25x - 48 = 0), solved using the quadratic formula with discriminant 817. The approximate solutions are (x \approx 1.8) and (x \approx -26.8). This algebraic process strengthens mathematical literacy and prepares you for complex problem-solving in advanced studies and real-life scenarios.", "---", "Keywords: quadratic equation, solve 4x² + 100x + 600 = 792, quadratic formula, solve x² + 25x - 48 = 0, approximate solutions, algebra, discriminant calculation, mathematical methods, high school math, quadratic solutions."]