2x^2 - 3x + 1 = 0

["# Solving the Quadratic Equation 2x² – 3x + 1 = 0: Step-by-Step Guide", "Mathematics provides powerful tools to solve real-world problems, and one of its essential applications lies in solving quadratic equations. The equation 2x² – 3x + 1 = 0 is a classic example that frequently appears in algebra, physics, and engineering. In this comprehensive article, we’ll explore how to solve this quadratic equation using multiple methods, understand its roots, and highlight practical uses in science and technology.", "---", "## 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 ≠ 0. The solutions to this equation represent the roots of the quadratic function ( f(x) = ax^2 + bx + c ), which graphically corresponds to a parabola intersecting the x-axis.", "---", "## Our Target Equation", "We focus on solving:", "$$\n2x^2 - 3x + 1 = 0\n$$", "Here,\n- ( a = 2 )\n- ( b = -3 )\n- ( c = 1 )", "---", "## Methods to Solve 2x² – 3x + 1 = 0", "There are three primary algebraic methods to solve quadratic equations:", "### 1. Factoring", "This method relies on expressing the quadratic expression as a product of two binomials.", "#### Step-by-step factoring:", "Start with\n$$\n2x^2 - 3x + 1\n$$", "We look for two numbers that multiply to ( a \cdot c = 2 \ imes 1 = 2 ) and add up to ( b = -3 ).", "The numbers –2 and –1 satisfy:\n(–2) × (–1) = 2\n(–2) + (–1) = –3", "We rewrite the middle term:\n$$\n2x^2 - 2x - x + 1 = 0\n$$", "Group the terms:\n$$\n(2x^2 - 2x) + (-x + 1) = 0\n$$", "Factor each group:\n$$\n2x(x - 1) -1(x - 1) = 0\n$$", "Factor out the common binomial:\n$$\n(x - 1)(2x - 1) = 0\n$$", "#### Solve for x:", "Set each factor equal to zero:\n- ( x - 1 = 0 ) → ( x = 1 )\n- ( 2x - 1 = 0 ) → ( x = \frac{1}{2} )", "✅ Roots: ( x = 1 ) and ( x = \frac{1}{2} )", "---", "### 2. Quadratic Formula", "For any quadratic equation ( ax^2 + bx + c = 0 ), the solutions are given by:", "$$\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n$$", "Plug in the values:\n( a = 2 ), ( b = -3 ), ( c = 1 )", "#### Compute the discriminant ( D = b^2 - 4ac ):\n$$\nD = (-3)^2 - 4(2)(1) = 9 - 8 = 1\n$$", "Since ( D > 0 ), there are two distinct real roots.", "#### Plug into the formula:\n$$\nx = \frac{-(-3) \pm \sqrt{1}}{2 \cdot 2} = \frac{3 \pm 1}{4}\n$$", "- ( x = \frac{3 + 1}{4} = \frac{4}{4} = 1 )\n- ( x = \frac{3 - 1}{4} = \frac{2}{4} = \frac{1}{2} )", "✅ Confirmed: roots are ( x = 1 ) and ( x = \frac{1}{2} )", "---", "### 3. Completing the Square (Alternative method)", "This geometric method converts the quadratic into a perfect square trinomial.", "Start with:\n$$\n2x^2 - 3x + 1 = 0\n$$", "#### Step 1: Move constant to the right:\n$$\n2x^2 - 3x = -1\n$$", "#### Step 2: Divide by coefficient of ( x^2 ) (2):\n$$\nx^2 - \frac{3}{2}x = -\frac{1}{2}\n$$", "#### Step 3: Complete the square:\nTake half of (-\frac{3}{2}): (-\frac{3}{4}), square it: ( \left(-\frac{3}{4}\right)^2 = \frac{9}{16} )", "Add to both sides:\n$$\nx^2 - \frac{3}{2}x + \frac{9}{16} = -\frac{1}{2} + \frac{9}{16}\n$$", "Convert right-hand side:\n[\n-\frac{8}{16} + \frac{9}{16} = \frac{1}{16}\n]", "Now left side is a perfect square:\n$$\n\left(x - \frac{3}{4}\right)^2 = \frac{1}{16}\n$$", "#### Step 4: Solve using square roots:\n[\nx - \frac{3}{4} = \pm \frac{1}{4}\n]", "So:\n- ( x = \frac{3}{4} + \frac{1}{4} = 1 )\n- ( x = \frac{3}{4} - \frac{1}{4} = \frac{1}{2} )", "✅ Roots are again confirmed: ( x = 1 ) and ( x = \frac{1}{2} )", "---", "## Why Is This Equation Important?", "The quadratic equation 2x² – 3x + 1 = 0 serves as a gateway to deeper mathematical concepts:", "- Physics: Modeling projectile motion where height depends quadratically on time.\n- Engineering: Calculating optimal points in parabolic reflectors or antennas.\n- Economics: Analyzing cost and revenue curves to determine break-even points.", "Understanding how to isolate real roots helps students build intuition for more complex systems and real-world modeling.", "---", "## Final Answer", "The solutions to the equation 2x² – 3x + 1 = 0 are:", "$$\nx = 1 \quad \ ext{and} \quad x = \frac{1}{2}\n$$", "Whether approached through factoring, the quadratic formula, or completing the square, each method validates the roots and enhances problem-solving flexibility.", "---", "## Key Takeaways\n- Factoring is fastest when a clear binomial pair exists (here, ( (2x - 1)(x - 1) )).\n- The discriminant ( D = 1 ) ensures two distinct real roots.\n- Applications extend beyond math—useful in science and optimization.", "---", "## Need More Practice? Try Solving Other Quadratics!", "Try solving these similar equations:\n- ( x^2 - 5x + 6 = 0 )\n- ( 3x^2 + 2x - 8 = 0 )", "Use factoring, the quadratic formula, or completing the square—each method offers valuable insight.", "---", "Keywords:\nquadratic equation solution, 2x² – 3x + 1 = 0, factoring quadratic, quadratic formula, solve quadratic equation, math tutorial, real roots, algebra 2, quadratic roots, discriminant analysis, solve 2x² – 3x + 1, step-by-step quadratic, quadratic functions, real-world applications of quadratics.", "Meta Description:\nSolve 2x² – 3x + 1 = 0 using factoring, quadratic formula, and completing the square. Learn real solutions, roots, and practical uses in science and engineering with clear, step-by-step explanations."]