Expanding, 300 + 70w + 4w² = 396.

["Solving the Equation 300 + 70w + 4w² = 396: A Step-by-Step Guide", "Crafting and solving quadratic equations is essential in algebra, and understanding how to expand, simplify, and solve equations like ( 300 + 70w + 4w² = 396 ) is a key skill for students, educators, and math enthusiasts. In this article, we’ll explore how to expand, rearrange, and solve this equation to find the value(s) of ( w ), providing clear, step-by-step guidance for anyone looking to master quadratic equations.", "---", "### Step 1: Rewrite the Equation in Standard Form", "The equation given is:", "[\n300 + 70w + 4w^2 = 396\n]", "To solve this, first bring all terms to one side and set the equation equal to zero:", "[\n4w^2 + 70w + 300 - 396 = 0\n]", "Simplify the constant terms:", "[\n4w^2 + 70w - 96 = 0\n]", "Now, the equation is in standard quadratic form:", "[\n4w^2 + 70w - 96 = 0\n]", "---", "### Step 2: Simplify the Equation (if possible)", "Before solving, simplify the equation by dividing every term by the greatest common divisor (GCD) of the coefficients. Here, 4 is the largest common factor:", "[\n\frac{4w^2}{4} + \frac{70w}{4} + \frac{−96}{4} = 0\n]", "[\nw^2 + 17.5w - 24 = 0\n]", "Though simplified, note that decimals (like 17.5) can be avoided by multiplying earlier. Instead of dividing by 4, work with the original simplified coefficients:", "[\n4w^2 + 70w - 96 = 0\n]", "This avoids rounding errors and keeps values precise.", "---", "### Step 3: Apply the Quadratic Formula", "The quadratic equation ( aw^2 + bw + c = 0 ) has the standard solution:", "[\nw = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "From our equation:", "- ( a = 4 )\n- ( b = 70 )\n- ( c = -96 )", "Plug values into the formula:", "[\nw = \frac{-70 \pm \sqrt{70^2 - 4 \cdot 4 \cdot (-96)}}{2 \cdot 4}\n]", "Calculate inside the square root:", "[\n70^2 = 4900\n]\n[\n4 \cdot 4 \cdot 96 = 1536\n]\n[\nb^2 - 4ac = 4900 + 1536 = 6436\n]", "So,", "[\nw = \frac{-70 \pm \sqrt{6436}}{8}\n]", "Now compute ( \sqrt{6436} ). Since ( 80^2 = 6400 ) and ( 81^2 = 6561 ), ( \sqrt{6436} ) is approximately 80.22.", "[\nw = \frac{-70 \pm 80.22}{8}\n]", "Compute both solutions:", "First solution:", "[\nw = \frac{-70 + 80.22}{8} = \frac{10.22}{8} \approx 1.2775\n]", "Second solution:", "[\nw = \frac{-70 - 80.22}{8} = \frac{-150.22}{8} \approx -18.7775\n]", "---", "### Step 4: Final Answer", "The equation ( 300 + 70w + 4w^2 = 396 ) expands and simplifies to the quadratic equation ( 4w^2 + 70w - 96 = 0 ), solved using the quadratic formula to yield approximate solutions:", "[\nw \approx 1.28 \quad \ ext{and} \quad w \approx -18.78\n]", "---", "### Why This Matters", "Solving quadratic equations like this one appears in physics, engineering, economics, and computer science. Understanding how to expand, simplify, and apply the quadratic formula empowers problem-solving across disciplines. Whether you’re modeling motion, optimizing profits, or analyzing data trends, equations such as the one above form foundational tools.", "---", "### Summary", "- Move all terms to one side: ( 4w^2 + 70w - 96 = 0 )\n- Apply the quadratic formula:\n [\n w = \frac{-70 \pm \sqrt{6436}}{8}\n ]\n- Approximate solutions:\n [\n w \approx 1.28 \quad \ ext{and} \quad w \approx -18.78\n ]\n- Easy-to-follow algebra enables stronger math skills and real-world applications.", "---", "Want to master quadratic equations? Practice simplifying, expanding, and solving various forms—every step builds confidence and clarity in algebra.", "---", "Keywords for SEO:\nExpanding quadratic equations, solve 4w² + 70w = 396, quadratic equation solution, simplify 300 + 70w + 4w² = 396, step-by-step quadratic, quadratic formula application, algebra problem solving, 300 + 70w + 4w² = 396\n---", "Start solving your quadratic equations confidently—every equation unlocks new understanding!"]