First, f(1) = 2(1)² − 3(1) + 1 = 2 − 3 + 1 = 0.

["# Understanding the First Value: f(1) = 2(1)² − 3(1) + 1 = 0 – A Complete Breakdown", "In mathematics, evaluating a function at a specific input is fundamental to understanding its behavior and purpose. One of the most straightforward examples is calculating the first value of a quadratic function, such as:", "[ f(1) = 2(1)^2 - 3(1) + 1 ]", "At first glance, this expression may seem simple, but computing it step-by-step reveals important principles of algebra and computation. This article explores how to evaluate ( f(1) ), why it equals zero, and the significance of this result.", "## Step-by-Step Calculation of f(1)", "### Step 1: Substitute ( x = 1 ) into the function\nGiven the function:\n[ f(x) = 2x^2 - 3x + 1 ]\nWe substitute ( x = 1 ):\n[\nf(1) = 2(1)^2 - 3(1) + 1\n]", "### Step 2: Apply exponentiation\nCalculate the square:\n[ (1)^2 = 1 ]\nNow multiply by the coefficient:\n[ 2 \ imes 1 = 2 ]", "### Step 3: Evaluate linear term\nMultiply the next term:\n[ -3 \ imes 1 = -3 ]", "### Step 4: Combine all terms\nPut the results together:\n[\nf(1) = 2 - 3 + 1\n]\nNow compute left to right:\n[ 2 - 3 = -1 ]\n[ -1 + 1 = 0 ]", "Thus,\n[ f(1) = 0 ]", "## Why Does f(1) = 0? The Mathematical Insight", "This result means that ( x = 1 ) is a root of the function ( f(x) ). More precisely, it is a root of multiplicity one, meaning the function crosses or touches the x-axis at ( x = 1 ).", "Evaluating functions at zero tells us whether a value is an input that satisfies the equation ( f(x) = 0 ), known as a solution or zero of the function. Here, substituting 1 gives zero — confirming that 1 is a precise solution to the equation ( f(x) = 0 ).", "## Significance of f(1) = 0 in Algebra", "1. Finding Roots\n Determining values like ( f(1) = 0 ) helps identify roots of quadratic equations, essential for factoring and solving higher-degree polynomials.", "2. Graph Behavior\n When a function evaluates to zero at ( x = 1 ), the graph of ( f(x) ) intersects the x-axis there, indicating a real, finite solution to ( f(x) = 0 ).", "3. Simplification of Expressions\n Knowing roots reduces complex polynomial expressions into factored forms, e.g., ( f(x) = (x - 1)(2x - 1) ), which simplifies solving, differentiation, and integration.", "## Real-World Applications of Zero Roots", "In physics and engineering, functions modeling behavior often yield zero values at critical points. For example, if ( f(x) ) represents a force or displacement, ( f(1) = 0 ) might indicate equilibrium or zero displacement at time ( t = 1 ).", "## Conclusion", "Calculating ( f(1) = 2(1)^2 - 3(1) + 1 ) is more than a computational exercise — it’s a gateway to understanding roots, graphing functions, and solving equations. With a clear step-by-step evaluation showing ( f(1) = 0 ), we confirm ( x = 1 ) as a fundamental solution, vital across algebra, calculus, and applied sciences. Remember: when ( f(a) = 0 ), ( a ) is a key solution — a building block for deeper mathematical insight.", "---", "Keywords: f(1), evaluating functions, quadratic function, root of a function, algebra, mathematical evaluation, zero of a function, polynomial roots\nMeta Description: Discover how evaluating f(1) = 2(1)² − 3(1) + 1 = 0 reveals a key root, fundamental in algebra and calculus. Learn step-by-step computation and real-world significance."]