\[ 2x^2 - 3x + 1 = 0. \] - United Radiology

["# Solving the Quadratic Equation: ( 2x^2 - 3x + 1 = 0 )", "Understanding how to solve quadratic equations is essential for learners and professionals in mathematics, physics, engineering, and computer science. One common example is the equation:", "[
\n2x^2 - 3x + 1 = 0
\n]", "In this comprehensive guide, we’ll explore how to solve this quadratic equation step-by-step, explain its real-world applications, and highlight important mathematical concepts. Whether you're studying algebra or preparing for exams, mastering this problem will strengthen your quadratic solving skills.", "---", "## What Is a Quadratic Equation?", "A quadratic equation is any equation of the standard form:", "[
\nax^2 + bx + c = 0
\n]", "where ( a ), ( b ), and ( c ) are constants and ( a <br/>\neq 0 ). The equation ( 2x^2 - 3x + 1 = 0 ) fits this definition with ( a = 2 ), ( b = -3 ), and ( c = 1 ).", "---", "## Why Solve ( 2x^2 - 3x + 1 = 0 )?", "Finding the roots (solutions) of this equation helps determine key points such as where a parabola crosses the x-axis. It also serves as a foundation for more advanced topics like optimization, motion modeling, and quadratic inequalities.", "---", "## Step-by-Step Solution Using the Quadratic Formula", "The most reliable method to solve any quadratic equation is using the quadratic formula:", "[
\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\n]", "### Step 1: Identify coefficients", "From ( 2x^2 - 3x + 1 = 0 ), extract:", "- ( a = 2 )
\n- ( b = -3 )
\n- ( c = 1 )", "### Step 2: Calculate the discriminant", "The discriminant ( D ) determines the nature and number of solutions:", "[
\nD = b^2 - 4ac = (-3)^2 - 4(2)(1) = 9 - 8 = 1
\n]", "Since ( D = 1 > 0 ), there are two distinct real roots.", "### Step 3: Apply the quadratic formula", "[
\nx = \frac{-(-3) \pm \sqrt{1}}{2(2)} = \frac{3 \pm 1}{4}
\n]", "Now compute both solutions:", "- ( x_1 = \frac{3 + 1}{4} = \frac{4}{4} = 1 )
\n- ( x_2 = \frac{3 - 1}{4} = \frac{2}{4} = \frac{1}{2} )", "---", "## Final Answer", "The solutions to the equation ( 2x^2 - 3x + 1 = 0 ) are:", "[
\nx = 1 \quad \ ext{and} \quad x = \frac{1}{2}
\n]", "These roots represent the x-intercepts of the corresponding parabola and are critical for graphing and analyzing quadratic functions.", "---", "## Real-World Applications", "Quadratic equations like this appear in diverse fields:", "- Physics: Modeling projectile motion under gravity.
\n- Engineering: Designing parabolic antennas and springs.
\n- Economics: Optimizing profit or cost functions.
\n- Computer Graphics: Rendering smooth curves and animations.", "Understanding the exact solutions enables precise predictions and optimizations.", "---", "## Summary", "- The equation ( 2x^2 - 3x + 1 = 0 ) is a standard quadratic equation.
\n- It can be solved using the quadratic formula with ( a = 2 ), ( b = -3 ), ( c = 1 ).
\n- The discriminant ( D = 1 ) confirms two distinct real roots: ( x = 1 ) and ( x = \frac{1}{2} ).
\n- Mastering this equation enhances problem-solving skills essential in science and engineering.", "---", "## Practice Problem Recap", "Solve:
\n[
\n2x^2 - 3x + 1 = 0
\n]", "Solutions:
\n[
\nx = 1 \quad \ ext{and} \quad x = \frac{1}{2}
\n]", "Roots:
\n.forEach(item =>
\n console.log(Root: ${item});
\n)
\n// Output:
\n// Root: 1
\n// Root: 0.5", "---", "### Further Reading & Resources", "- Quadratic Formula Tutorial
\n- Graphing Parabolas
\n- Quadratic Inequalities Explained", "---", "Understanding and solving ( 2x^2 - 3x + 1 = 0 ) equips you with foundational tools for advanced mathematics. Keep practicing, and explore how quadratics shape technology and science!"]