2x^2 - 2x + 3x - 3 = 40 \Rightarrow 2x^2 + x - 43 = 0

["# Solve the Quadratic Equation: ( 2x^2 + x - 43 = 0 )", "Quadratic equations are fundamental in algebra and appear frequently in various fields such as physics, engineering, and economics. One such equation that often arises in problem-solving scenarios is:", "[\n2x^2 + x - 43 = 0\n]", "But how do we solve this equation? In this article, we’ll walk through the step-by-step process of solving ( 2x^2 + x - 43 = 0 ), why it simplifies to forms often encountered like ( 2x^2 - 2x + 3x - 3 = 40 ), and provide tips for tackling similar quadratic problems.", "---", "## Understanding the Equation: Rearranging and Simplifying", "Our target equation is", "[\n2x^2 + x - 43 = 0\n]", "At first glance, it appears straightforward, but sometimes expressions are rearranged or simplified before solving. For instance, you may encounter a form like:", "[\n2x^2 - 2x + 3x - 3 = 40\n]", "This equation is equivalent to the original because:", "[\n2x^2 + x - 43 = 0 \Rightarrow 2x^2 + x = 43\n]", "Let’s combine terms:", "[\n2x^2 + x = 43 \quad \ ext{so} \quad 2x^2 + x - 43 = 0\n]", "Notice:", "- ( -2x + 3x = x )\n- Rearranging terms gives a form that might be used to factor or complete the square.", "While this form isn’t always necessary for solving, understanding transformations helps when equations are presented in alternative ways.", "---", "## Applying the Quadratic Formula", "For any quadratic equation in the standard form ( ax^2 + bx + c = 0 ), the quadratic formula provides the exact solutions:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "For our equation:", "[\n2x^2 + x - 43 = 0\n]", "Identify coefficients:", "- ( a = 2 )\n- ( b = 1 )\n- ( c = -43 )", "Plug into the formula:", "[\nx = \frac{-1 \pm \sqrt{(1)^2 - 4(2)(-43)}}{2(2)} = \frac{-1 \pm \sqrt{1 + 344}}{4} = \frac{-1 \pm \sqrt{345}}{4}\n]", "---", "## Why is This Useful?", "The exact solution is expressed in simplified radical form. While decimal approximations are tempting, symbolic representation using ( \sqrt{345} ) preserves precision.", "245345≈18.58:", "[\nx \approx \frac{-1 + 18.58}{4} \approx \frac{17.58}{4} \approx 4.395\n]", "[\nx \approx \frac{-1 - 18.58}{4} = \frac{-19.58}{4} \approx -4.895\n]", "So the two real solutions are approximately:", "[\nx \approx 4.40 \quad \ ext{and} \quad x \approx -4.90\n]", "---", "## Real-World Applications", "Quadratic equations model real-life phenomena:", "- Projectile motion: The height of a launched object follows a quadratic path.\n- Profit maximization: Revenue and cost functions often form quadratics.\n- Optimization: Finding maximum or minimum values in economics and engineering.", "---", "## Tips for Solving Quadratics", "- Rewrite equations carefully: Combine like terms before applying formulas.\n- Check discriminant: ( b^2 - 4ac ) determines the nature of roots (real, repeated, or imaginary).\n- Use factoring if possible: Sometimes expressions simplify neatly into factors.\n- Verify solutions: Substitute back into the original equation to ensure correctness.", "---", "## Conclusion", "Solving ( 2x^2 + x - 43 = 0 ) illustrates key algebraic techniques involving coefficients, the quadratic formula, and simplified expressions. Though alternative forms like ( 2x^2 - 2x + 3x - 3 = 40 ) may arise, the core method remains consistent: identify ( a ), ( b ), and ( c ), apply the quadratic formula, and interpret the results.", "Whether you're tackling equations on homework, by hand, or with a calculator, mastering quadratic solutions empowers deeper problem-solving across disciplines.", "---", "Key Takeaway:\nTo solve ( 2x^2 + x - 43 = 0 ), use:", "[\nx = \frac{-1 \pm \sqrt{345}}{4}\n]", "This approach ensures accuracy and deepens your understanding of quadratic behavior.", "---", "Want more algebra tips? Subscribe to our newsletter for weekly problem-solving guides and deep dives into quadratic equations and beyond!"]