Simplifying: \( 4x^2 + 100x + 600 = 936 \).
["# Simplifying the Equation: ( 4x^2 + 100x + 600 = 936 )", "Solving quadratic equations can seem complicated at first, but with a systematic approach, even complex equations like ( 4x^2 + 100x + 600 = 936 ) become much easier to manage. This article walks you through the step-by-step process of simplifying and solving this equation, helping you build confidence in handling similar quadratic expressions.", "---", "## Step 1: Move All Terms to One Side", "To simplify, begin by bringing all terms to one side of the equation to form a standard quadratic equation equal to zero:", "[\n4x^2 + 100x + 600 - 936 = 0\n]", "Simplify the constants:", "[\n4x^2 + 100x - 336 = 0\n]", "This is now in the form:", "[\n\boxed{4x^2 + 100x - 336 = 0}\n]", "---", "## Step 2: Simplify the Equation (Optional)", "To reduce complexity, divide every term by the greatest common divisor (GCD) of the coefficients. Here, the GCD of 4, 100, and 336 is 4:", "[\n\frac{4x^2}{4} + \frac{100x}{4} - \frac{336}{4} = 0\n]", "Which simplifies to:", "[\nx^2 + 25x - 84 = 0\n]", "Now you’re working with a simpler, equivalent equation.", "---", "## Step 3: Solve the Quadratic Equation", "Use the quadratic formula to solve ( x^2 + 25x - 84 = 0 ):", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Here, ( a = 1 ), ( b = 25 ), and ( c = -84 ).", "Calculate the discriminant:", "[\n\Delta = 25^2 - 4(1)(-84) = 625 + 336 = 961\n]", "Now compute:", "[\nx = \frac{-25 \pm \sqrt{961}}{2} = \frac{-25 \pm 31}{2}\n]", "So:", "- ( x = \frac{-25 + 31}{2} = \frac{6}{2} = 3 )\n- ( x = \frac{-25 - 31}{2} = \frac{-56}{2} = -28 )", "---", "## Step 4: Final Answer", "The solutions to the equation ( 4x^2 + 100x + 600 = 936 ) are:", "[\n\boxed{x = 3 \quad \ ext{and} \quad x = -28}\n]", "---", "## Why Simplifying Helps", "- Reduces errors by working with simpler coefficients.\n- Makes applying the quadratic formula straightforward.\n- Helps identify key features like vertex and intercepts more easily.", "---", "## Full Simplified Equation Summary", "[\n4x^2 + 100x - 336 = 0 \quad \ ext{or} \quad x^2 + 25x - 84 = 0\n]", "With distinct real solutions, this equation is fully simplified and ready for graphing or application.", "---", "Bonus Tip: Always verify your solutions by substituting them back into the original equation. Try plugging ( x = 3 ) and ( x = -28 ) into ( 4x^2 + 100x + 600 ) — both should yield 936.", "---", "Keywords: Simplify quadratic equation, solve ( 4x^2 + 100x + 600 = 936 ), step-by-step quadratic, quadratic formula, simplify algebraic expressions, solve ( x^2 + 25x - 84 = 0 ), quadratic solutions."]
