4x^2 + 50x + 150 = 231

["# How to Solve the Quadratic Equation: 4x² + 50x + 150 = 231 — Step-by-Step Guide", "If you’ve stumbled upon the equation 4x² + 50x + 150 = 231, you’re not alone — quadratic equations like this are common in algebra, physics, and real-world problem-solving. This article walks you through solving this equation step-by-step and explains how to interpret its solutions. Plus, we cover will how to find the roots, analyze graph behavior, and apply this in practical contexts — all optimized for SEO.", "## Understanding the Equation", "The given equation is a standard quadratic equation in the form:", "[\nax^2 + bx + c = 0\n]", "But first, let’s rewrite it into standard form by moving 231 to the left side:", "[\n4x^2 + 50x + 150 - 231 = 0\n]", "Simplifying,", "[\n4x^2 + 50x - 81 = 0\n]", "Now we have:", "- ( a = 4 )\n- ( b = 50 )\n- ( c = -81 )", "## Step 1: Using the Quadratic Formula", "The most reliable method to solve a quadratic equation is the quadratic formula:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Plugging in the values:", "[\nx = \frac{-50 \pm \sqrt{(50)^2 - 4 \cdot 4 \cdot (-81)}}{2 \cdot 4}\n]", "[\nx = \frac{-50 \pm \sqrt{2500 + 1296}}{8}\n]", "[\nx = \frac{-50 \pm \sqrt{3796}}{8}\n]", "Now calculate the square root of 3796. Using a calculator:", "[\n\sqrt{3796} \approx 61.62\n]", "So,", "[\nx = \frac{-50 \pm 61.62}{8}\n]", "### Calculating the Two Solutions", "First solution (positive root):", "[\nx = \frac{-50 + 61.62}{8} = \frac{11.62}{8} \approx 1.4525\n]", "Second solution (negative root):", "[\nx = \frac{-50 - 61.62}{8} = \frac{-111.62}{8} \approx -13.9525\n]", "---", "## Step 2: Verification Using Graphing", "You can confirm these solutions by graphing the function:", "[\ny = 4x^2 + 50x - 81\n]", "The graph is a parabola opening upward since ( a = 4 > 0 ). The roots appear at approximately ( x \approx -13.95 ) and ( x \approx 1.45 )—matching our algebraic solution.", "---", "## Step 3: Real-World Application", "Equations like 4x² + 50x + 150 = 231 often arise in:", "- Physics: Calculating motion, projectile trajectories, or energy calculations.\n- Engineering: Modeling costs, structural loads, or signal processing.\n- Business: Profit maximization, break-even analysis.", "For example, if ( x ) represents time, the equation models when a system reaches a target output, where 231 is a threshold value.", "---", "## Final Thoughts and Key Takeaways", "- Always rewrite the equation in standard form before applying formulas.\n- The discriminant ( b^2 - 4ac ) determines the nature of roots (real, repeated, or imaginary).\n- Use the quadratic formula for accuracy and verification.\n- Use graphing tools to visualize solutions and confirm algebraically derived answers.", "Understanding how to solve 4x² + 50x + 150 = 231 empowers you to tackle similar quadratic problems confidently—whether in academics, exams, or real-life applications.", "---", "## Related SEO Keywords", "- how to solve 4x² + 50x + 150 = 231\n- solve quadratic equations step-by-step\n- find roots of ax² + bx + c = 0\n- quadratic formula tutorial\n- analyze 4x² + 50x – 81 = 0\n- real-world applications of quadratic equations", "---", "Optimize your learning and solving skills today — mastering quadratics starts here!\nIf you enjoyed this guide, share it with classmates, bookmark it for future reference, and explore more advanced algebra topics such as completing the square and quadratic functions.", "---", "Keywords: 4x² + 50x + 150 = 231, solve quadratic equations, quadratic formula, algebraic solutions, real-world applications, graphing quadratic functions, step-by-step solving"]