Expand: \(250 + 50x + 20x + 4x^2 = 600\).
["Expand and Solve: Mastering the Expansion of (250 + 50x + 20x + 4x^2 = 600)", "In algebra, simplifying and solving equations step-by-step is a crucial skill that opens the door to deeper understanding and problem-solving. One common exercise involves expanding and simplifying polynomial equations, which helps clarify their structure and rearrange them into standard forms for easier analysis.", "Today, we explore the equation:", "[\n250 + 50x + 20x + 4x^2 = 600\n]", "### Step 1: Combine Like Terms", "Before expanding (the equation is already partially combined), simplify the left-hand side by combining like terms:", "[\n50x + 20x = 70x\n]", "So the equation becomes:", "[\n4x^2 + 70x + 250 = 600\n]", "### Step 2: Rearrange to Standard Quadratic Form", "To solve for (x), bring all terms to one side to form a standard quadratic equation:", "[\n4x^2 + 70x + 250 - 600 = 0\n]", "[\n4x^2 + 70x - 350 = 0\n]", "### Step 3: Simplify (Optional)", "We can simplify the equation by dividing all terms by the greatest common factor, which is 2:", "[\n2x^2 + 35x - 175 = 0\n]", "This form makes solving easier using the quadratic formula or factoring.", "### Step 4: Solve the Quadratic Equation", "We apply the quadratic formula:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "For (2x^2 + 35x - 175 = 0), coefficients are:\n- (a = 2)\n- (b = 35)\n- (c = -175)", "Calculate the discriminant:", "[\nb^2 - 4ac = 35^2 - 4(2)(-175) = 1225 + 1400 = 2625\n]", "Find square root of discriminant:", "[\n\sqrt{2625} = \sqrt{25 \cdot 105} = 5\sqrt{105}\n]", "Substitute into the formula:", "[\nx = \frac{-35 \pm 5\sqrt{105}}{4}\n]", "### Step 5: Final Answer", "Thus, the solutions to the equation (250 + 50x + 20x + 4x^2 = 600) are:", "[\nx = \frac{-35 + 5\sqrt{105}}{4} \quad \ ext{and} \quad x = \frac{-35 - 5\sqrt{105}}{4}\n]", "---", "### Why This Matters", "Understanding how to expand and simplify polynomial equations like this is foundational in algebra. It not only helps solve specific problems but also improves your ability to manipulate and analyze complex expressions—skills essential in higher mathematics, engineering, and data science applications.", "---", "### SEO Keywords:\nquadratic equation expansion, solve 2x² + 70x - 350 = 0, simplify 4x² + 70x + 250 = 600, algebra solving steps, polynomial equation solving, expand and solve quadratic equation", "---", "Mastering these techniques takes practice—but every expansion brings you closer to confident problem-solving. Keep practicing, and watch your algebra fluency grow!"]
