Expand: \( 300 + 40x + 30x + 4x^2 = 504 \)

["Title: Expand and Solve the Quadratic Equation: ( 300 + 40x + 30x + 4x^2 = 504 )", "---", "Meta Description:\nExpand the equation ( 300 + 40x + 30x + 4x^2 = 504 ), simplify it into standard quadratic form, and solve step-by-step—perfect for high school algebra and math learners.", "---", "### Introduction", "Solving quadratic equations is a fundamental skill in algebra, crucial for students, educators, and self-learners. One such equation students often encounter is:", "[\n300 + 40x + 30x + 4x^2 = 504\n]", "At first glance, this expression may seem complex, but expanding and simplifying it reveals a clear quadratic form that’s easy to solve. In this article, we’ll expand the expression step-by-step, combine like terms, rewrite the equation in standard form, and solve it using familiar techniques like factoring and the quadratic formula.", "---", "### Step 1: Combine Like Terms", "The left-hand side of the equation contains several terms with ( x ) and a constant. Let’s combine the linear (first-degree) terms:", "[\n40x + 30x = 70x\n]", "So the equation becomes:", "[\n4x^2 + 70x + 300 = 504\n]", "---", "### Step 2: Move All Terms to One Side (Standard Form)", "To write the equation in standard quadratic form ( ax^2 + bx + c = 0 ), move 504 to the left side:", "[\n4x^2 + 70x + 300 - 504 = 0\n]", "Simplify the constants:", "[\n4x^2 + 70x - 204 = 0\n]", "This is the simplified quadratic equation ready for solving.", "---", "### Step 3: Simplify the Equation (Optional But Helpful)", "To make calculations easier, divide every term by the greatest common divisor (GCD) of 4, 70, and 204.", "- GCD(4, 70, 204) = 2", "Divide each term by 2:", "[\n2x^2 + 35x - 102 = 0\n]", "此简化后的方程更易于处理，同时保持等式成立。", "---", "### Step 4: Solve the Quadratic Equation", "We now solve:", "[\n2x^2 + 35x - 102 = 0\n]", "#### Option A: Factoring (if possible)", "Check if the quadratic factors nicely. We look for two numbers that multiply to ( 2 \ imes (-102) = -204 ) and add up to ( 35 ).", "After testing possible factor pairs, we find:", "- ( 39 \ imes (-5.23) ) → not integers\n- ( 51 \ imes (-4) = -204 ), but ( 51 - 4 = 47 )\n- Try ( 39 \ imes (-5.23) ) doesn’t work", "No clean integer factors mean factoring is difficult here. Proceed with the quadratic formula.", "---", "#### Option B: Quadratic Formula", "The quadratic formula for ( ax^2 + bx + c = 0 ) is:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "From ( 2x^2 + 35x - 102 = 0 ), we identify:", "- ( a = 2 )\n- ( b = 35 )\n- ( c = -102 )", "Now compute the discriminant:", "[\n\Delta = b^2 - 4ac = 35^2 - 4(2)(-102) = 1225 + 816 = 2041\n]", "So:", "[\nx = \frac{-35 \pm \sqrt{2041}}{4}\n]", "Since ( \sqrt{2041} ) is not a perfect square, we keep the solution in radical form.", "---", "### Step 5: Approximate Solutions (Optional)", "Approximate ( \sqrt{2041} \approx 45.18 ):", "[\nx = \frac{-35 \pm 45.18}{4}\n]", "Calculate both roots:", "- ( x_1 = \frac{-35 + 45.18}{4} = \frac{10.18}{4} \approx 2.55 )\n- ( x_2 = \frac{-35 - 45.18}{4} = \frac{-80.18}{4} \approx -20.05 )", "---", "### Conclusion", "The expanded and simplified form of the equation\n[\n300 + 40x + 30x + 4x^2 = 504\n]\nis:", "[\n2x^2 + 35x - 102 = 0\n]", "Using the quadratic formula, the solutions are:", "[\nx = \frac{-35 \pm \sqrt{2041}}{4}\n]", "For exact solutions, leave in radical form; for approximations, ( x \approx 2.55 ) and ( x \approx -20.05 ).", "---", "### Key Takeaways", "- Combining like terms simplifies complex expressions.\n- Always rewrite equations to standard quadratic form.\n- When factoring is difficult, use the quadratic formula with discriminant analysis.\n- Practice solving similar equations to build fluency.", "---", "Keywords: expand quadratic equation, solve 4x² + 70x - 204 = 0, step-by-step algebra, quadratic formula, simplify polynomial, algebra practice, middle school math.", "---", "Additional Resources:\n- Khan Academy: Quadratic Equations\n- AlgebraHelp.com: Solving Quadratics\n- Math is Fun: Quadratic Equations", "---", "Ready to expand your algebra skills? Start solving equations like a pro with this step-by-step guide!"]