2x^2 + x - 1 = 0

["Solving the Quadratic Equation 2x² + x – 1 = 0: Step-by-Step Guide", "Solving quadratic equations is a fundamental skill in algebra, essential for students, engineers, and scientists alike. The equation 2x² + x – 1 = 0 is a classic example of a quadratic equation that can be solved using standard algebraic methods. In this article, we’ll explore how to solve this equation step-by-step using the quadratic formula, analyze its roots, and understand real-world applications.", "---", "### What Is a Quadratic Equation?", "A quadratic equation is any equation of the form:", "$$\nax^2 + bx + c = 0\n$$", "where ( a ), ( b ), and ( c ) are constants and ( a <br/>\ne 0 ). The equation 2x² + x – 1 = 0 fits this form with\n- ( a = 2 ),\n- ( b = 1 ),\n- ( c = -1 ).", "---", "### Methods to Solve 2x² + x – 1 = 0", "There are three primary methods for solving quadratics:\n1. Factoring\n2. Completing the square\n3. Quadratic Formula", "For this equation, factoring is possible, but completing the square or applying the quadratic formula offers reliable results.", "---", "### Method 1: Using the Quadratic Formula", "The quadratic formula provides a direct solution for any quadratic equation and is given by:", "$$\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n$$", "Substituting ( a = 2 ), ( b = 1 ), and ( c = -1 ):", "1. Calculate the discriminant:\n$$\n\Delta = b^2 - 4ac = 1^2 - 4(2)(-1) = 1 + 8 = 9\n$$", "2. Since the discriminant is positive, there are two real and distinct solutions.", "3. Plug values into the formula:", "$$\nx = \frac{-1 \pm \sqrt{9}}{2 \ imes 2} = \frac{-1 \pm 3}{4}\n$$", "4. Solve both possibilities:", "- ( x = \frac{-1 + 3}{4} = \frac{2}{4} = \frac{1}{2} )\n- ( x = \frac{-1 - 3}{4} = \frac{-4}{4} = -1 )", "Solutions:\n$$\nx = \frac{1}{2} \quad \ ext{and} \quad x = -1\n$$", "---", "### Method 2: Factoring (Verification)", "Let’s verify by factoring. We want two numbers that multiply to ( 2 \ imes (-1) = -2 ) and add to ( 1 ). The numbers ( 2 ) and ( -1 ) work.", "Rewrite the middle term:", "$$\n2x^2 + 2x - x - 1 = 0\n$$", "Group terms:", "$$\n(2x^2 + 2x) + (-x - 1) = 0\n=> 2x(x + 1) - 1(x + 1) = 0\n=> (2x - 1)(x + 1) = 0\n$$", "Set each factor to zero:", "- ( 2x - 1 = 0 \Rightarrow x = \frac{1}{2} )\n- ( x + 1 = 0 \Rightarrow x = -1 )", "Matches the quadratic formula results.", "---", "### Graphical Interpretation", "The quadratic function ( y = 2x^2 + x - 1 ) is a parabola opening upward (because ( a = 2 > 0 )). The roots ( x = -1 ) and ( x = \frac{1}{2} ) are the points where the graph intersects the x-axis.", "---", "### Real-World Applications", "Quadratic equations model various physical and economic phenomena, such as:", "- Projectile motion (e.g., predicting the height of a launched object)\n- Optimization problems (maximizing profit or minimizing cost)\n- Physics trajectories and engineering design", "Understanding how to solve equations like ( 2x^2 + x - 1 = 0 ) equips you with tools for analyzing these real-life scenarios.", "---", "### Summary", "- The equation 2x² + x – 1 = 0 yields solutions ( x = -1 ) and ( x = \frac{1}{2} ).\n- Solutions were found using the quadratic formula and verified by factoring.\n- The discriminant (( \Delta = 9 )) confirms two distinct real roots.\n- This kind of equation appears frequently in science, math, and engineering.", "Mastering quadratic equations not only boosts algebraic proficiency but also enhances problem-solving skills for practical applications.", "---", "Keywords:\nquadratic equation 2x² + x – 1 = 0, solving quadratic equations, quadratic formula, factoring, discriminant, real roots, algebra tutorial, quadratic solutions, math tips", "Meta Description:\nSolve 2x² + x – 1 = 0 step-by-step using the quadratic formula. Learn about real roots, graph interpretation, and applications in science and engineering."]