Wobei \( a = 2 \), \( b = 3 \), \( c = -2 \):
["# Understanding Wobei: Solving the Equation ( 2x^2 + 3x - 2 = 0 )", "When diving into algebraic equations, quadratic expressions often present both challenges and opportunities for learning. One such relevant example is solving the quadratic equation defined by:", "[ 2x^2 + 3x - 2 = 0 ]", "Setting ( a = 2 ), ( b = 3 ), and ( c = -2 ), this equation follows the standard form ( ax^2 + bx + c = 0 ). Solving it not only sharpens algebra skills but also forms the foundation for analyzing real-world problems in fields like physics, engineering, and economics.", "## The Quadratic Formula: A Powerful Solution Tool", "For quadratic equations, the quadratic formula provides a direct and reliable method to find roots:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Plugging in ( a = 2 ), ( b = 3 ), ( c = -2 ):", "[\nx = \frac{-3 \pm \sqrt{(3)^2 - 4(2)(-2)}}{2(2)} = \frac{-3 \pm \sqrt{9 + 16}}{4} = \frac{-3 \pm \sqrt{25}}{4}\n]", "Since ( \sqrt{25} = 5 ), we simplify:", "[\nx = \frac{-3 \pm 5 }{4}\n]", "Now compute both roots:", "- ( x = \frac{-3 + 5}{4} = \frac{2}{4} = \frac{1}{2} )\n- ( x = \frac{-3 - 5}{4} = \frac{-8}{4} = -2 )", "Thus, the solutions are ( x = \frac{1}{2} ) and ( x = -2 ).", "## Verifying the Roots: Ensuring Accuracy", "To confirm correctness, substitute each value back into the original equation:", "For ( x = \frac{1}{2} ):\n[\n2\left(\frac{1}{2}\right)^2 + 3\left(\frac{1}{2}\right) - 2 = 2\left(\frac{1}{4}\right) + \frac{3}{2} - 2 = \frac{1}{2} + \frac{3}{2} - 2 = 2 - 2 = 0\n]", "For ( x = -2 ):\n[\n2(-2)^2 + 3(-2) - 2 = 2(4) - 6 - 2 = 8 - 6 - 2 = 0\n]", "Both roots satisfy the equation, verifying their validity.", "## Applications of the Solutions", "Understanding the roots of ( 2x^2 + 3x - 2 = 0 ) has practical relevance. In physics, such equations model projectile motion under gravity. In economics, they might represent break-even points where cost equals revenue. Exploring these real-world connections helps turn abstract algebra into tangible insights.", "## Step-by-Step Summary", "| Step | Action | Result |\n|-------|--------|--------|\n| 1 | Identify coefficients: ( a = 2 ), ( b = 3 ), ( c = -2 ) | Confirmed input values |\n| 2 | Apply quadratic formula: ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ) | Formula setup complete |\n| 3 | Calculate discriminant: ( b^2 - 4ac = 25 ) | Discriminant positive → two real roots |\n| 4 | Solve for ( x ): roots are ( \frac{1}{2} ) and ( -2 ) | Verified via substitution |\n| 5 | Real-world applications | Motion, economics; deeper understanding |", "## Conclusion", "Solving ( 2x^2 + 3x - 2 = 0 ) exemplifies how algebraic methods like the quadratic formula enable precise problem-solving. With ( a = 2 ), ( b = 3 ), and ( c = -2 ), the equation yields roots ( x = \frac{1}{2} ) and ( x = -2 )—solutions that not only satisfy the equation but also unlock applications across science and industry. Mastering such techniques empowers learners to tackle complex challenges with confidence."]
