Solve using the quadratic formula: \(x = \frac{-70 \pm \sqrt{4900 + 5600}}{8}\).

["# Solve Using the Quadratic Formula: Solve (x = \frac{-70 \pm \sqrt{4900 + 5600}}{8})", "When solving quadratic equations, the quadratic formula is one of the most powerful tools in algebra. In this article, we’ll walk through solving the equation (x = \frac{-70 \pm \sqrt{4900 + 5600}}{8}) step by step using the quadratic formula, helping you understand not just the solution but the underlying method.", "---", "## Understanding the Quadratic Formula", "The general form of a quadratic equation is:", "[\nax^2 + bx + c = 0\n]", "The quadratic formula gives the solutions for (x):", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "When it’s not immediately factored, rewriting the equation into standard form enables us to apply this formula directly.", "---", "## Step 1: Identify Coefficients", "Given the expression:", "[\nx = \frac{-70 \pm \sqrt{4900 + 5600}}{8}\n]", "First, simplify the discriminant:", "[\n4900 + 5600 = 10500\n]", "So the equation becomes:", "[\nx = \frac{-70 \pm \sqrt{10500}}{8}\n]", "However, to use the quadratic formula cleanly and solve explicitly for (x), we can reframe this into a standard quadratic equation. Notice that:", "[\nx = \frac{-70}{8} \pm \frac{\sqrt{10500}}{8}\n]", "But since the inequality under the radical sums two numbers, we interpret the numerator as arising from:", "[\nx = \frac{-70 \pm \sqrt{4900 + 5600}}{8} = \frac{-70 \pm \sqrt{10500}}{8}\n]", "This directly corresponds to a quadratic equation whose solution is implied by the quadratic formula.", "---", "## Step 2: Rewrite as a Quadratic Equation", "Let’s derive the standard form. Starting from:", "[\nx = \frac{-70 \pm \sqrt{10500}}{8}\n]", "Multiply both sides by 8:", "[\n8x = -70 \pm \sqrt{10500}\n]", "Rearranged:", "[\n8x + 70 = \pm \sqrt{10500}\n]", "Now square both sides to eliminate the square root:", "[\n(8x + 70)^2 = 10500\n]", "Expanding the left side:", "[\n64x^2 + 1120x + 4900 = 10500\n]", "Bring all terms to one side:", "[\n64x^2 + 1120x + 4900 - 10500 = 0\n]", "[\n64x^2 + 1120x - 5600 = 0\n]", "Now divide the entire equation by 16 to simplify coefficients:", "[\n4x^2 + 70x - 350 = 0\n]", "This is the quadratic equation whose solutions are given by:", "[\nx = \frac{-70 \pm \sqrt{4900 + 5600}}{8}\n]", "as originally stated. Check discriminant consistency:", "[\nb^2 - 4ac = 70^2 - 4(4)(-350) = 4900 + 5600 = 10500\n]", "Perfect — matches the discriminant used.", "---", "## Step 3: Apply the Quadratic Formula", "Now apply:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-70 \pm \sqrt{4900 + 5600}}{8}\n]", "Calculate the square root:", "[\n\sqrt{10500} \approx 102.4695\n]", "So two solutions:", "[\nx = \frac{-70 + 102.4695}{8} \approx \frac{32.4695}{8} \approx 4.0557\n]", "[\nx = \frac{-70 - 102.4695}{8} \approx \frac{-172.4695}{8} \approx -21.5587\n]", "---", "## Conclusion: Solutions and Insight", "The solutions to the equation (x = \frac{-70 \pm \sqrt{4900 + 5600}}{8}) are:", "[\nx = \frac{-70 \pm \sqrt{10500}}{8} \approx 4.06 \quad \ ext{and} \quad x \approx -21.56\n]", "More importantly, this expression leads naturally to the standard quadratic equation (4x^2 + 70x - 350 = 0), showing how the quadratic formula is derived from completing the square or direct algebraic manipulation.", "---", "## Why This Matters", "Using the quadratic formula avoids messy factoring and provides exact solutions even when roots are irrational. Whether solving for physics problems, economic models, or geometry, this method is consistent and reliable.", "Try solving your next quadratic by first ensuring it’s in standard form, then apply the formula confidently.", "---", "## Key Takeaways", "- Always simplify the discriminant before plugging into the formula.\n- The expression (\sqrt{b^2 - 4ac}) may include large numbers — factoring common terms helps.\n- Verify solutions by substitution or graphing.\n- Mastering this formula strengthens algebraic fluency across STEM fields.", "---", "Keywords: quadratic equation, solve using quadratic formula, quadratic formula steps, solve (x = \frac{-70 \pm \sqrt{4900 + 5600}}{8}), discriminant calculation, algebraic methods, solving quadratics with square roots.", "---", "Meta Description:\nLearn how to solve (x = \frac{-70 \pm \sqrt{4900 + 5600}}{8}) using the quadratic formula. Step-by-step explanation, simplification, and verified solutions included. Perfect for students and math learners."]