Expanding: \( 600 + 60x + 40x + 4x^2 = 936 \).

["# Solving the Quadratic Equation: ( 600 + 60x + 40x + 4x^2 = 936 )", "Expanding and simplifying algebraic expressions is a fundamental skill in mathematics, essential for solving equations in real-world applications. One such problem is expanding and solving the equation:", "[\n600 + 60x + 40x + 4x^2 = 936\n]", "This article guides you step-by-step through simplifying the equation, rewriting it in standard quadratic form, and solving it efficiently—key techniques that not only help find solutions but also improve your algebra mastery for advanced problem-solving.", "---", "## Step 1: Simplify the Left-Hand Side", "Begin by combining like terms on the left-hand side. Notice that ( 60x + 40x ) can be combined:", "[\n600 + (60x + 40x) + 4x^2 = 936\n]\n[\n600 + 100x + 4x^2 = 936\n]", "Now the equation is:", "[\n4x^2 + 100x + 600 = 936\n]", "---", "## Step 2: Rewrite in Standard Quadratic Form", "To solve for ( x ), bring all terms to one side so the right-hand side becomes zero:", "[\n4x^2 + 100x + 600 - 936 = 0\n]\n[\n4x^2 + 100x - 336 = 0\n]", "This is now in the standard quadratic form:", "[\nax^2 + bx + c = 0 \quad \ ext{where} \quad a = 4,\ b = 100,\ c = -336\n]", "---", "## Step 3: Simplify the Equation (Optional but Useful)", "Before applying the quadratic formula, simplify the equation by dividing all terms by the greatest common divisor (GCD) of coefficients ( 4, 100, ) and ( 336 ). The GCD is 4:", "[\n\frac{4x^2}{4} + \frac{100x}{4} + \frac{-336}{4} = 0\n]\n[\nx^2 + 25x - 84 = 0\n]", "Now you face a simpler quadratic:", "[\nx^2 + 25x - 84 = 0\n]", "---", "## Step 4: Solve Using the Quadratic Formula", "Since factoring may not always be straightforward, use the quadratic formula:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Here, ( a = 1 ), ( b = 25 ), ( c = -84 ). Plug in:", "[\nx = \frac{-25 \pm \sqrt{25^2 - 4(1)(-84)}}{2(1)}\n]\n[\nx = \frac{-25 \pm \sqrt{625 + 336}}{2}\n]\n[\nx = \frac{-25 \pm \sqrt{961}}{2}\n]\n[\nx = \frac{-25 \pm 31}{2}\n]", "This yields two solutions:", "1. ( x = \frac{-25 + 31}{2} = \frac{6}{2} = 3 )\n2. ( x = \frac{-25 - 31}{2} = \frac{-56}{2} = -28 )", "---", "## Step 5: Final Answer and Interpretation", "The solutions to the equation ( 600 + 60x + 40x + 4x^2 = 936 ) are:", "[\n\boxed{x = 3 \quad} \ ext{and} \quad \boxed{x = -28}\n]", "These solutions represent critical points—useful in modeling real-world scenarios such as break-even analysis, projectile motion, or cost optimization—where quadratic relationships model behavior.", "---", "## Why Practice Expanding and Solving Quadratic Equations?", "- Enhances algebraic fluency: Mastery of expanding and simplifying expressions enables tackling complex equations across physics, economics, and engineering.\n- Builds logical reasoning: Learning to transform, simplify, and solve equations strengthens structured thinking.\n- Prepares for advanced math: Foundational for calculus, linear algebra, and computational problem-solving.", "---", "### Summary", "Expanding ( 600 + 60x + 40x + 4x^2 ) leads directly to a quadratic form solvable via standard methods. Combining like terms, applying the quadratic formula, and simplifying yield two real solutions: ( x = 3 ) and ( x = -28 ). Whether studying algebra or applying it in real disciplines, understanding this process empowers accurate problem-solving and deeper mathematical insight.", "---", "Keywords: Quadratic equation, solve (4x^2 + 100x - 336 = 0), simplify algebra, quadratic formula, (600 + 60x + 40x + 4x^2 = 936) solutions, expand polynomial, algebra practice."]