f(3) = 2(3)^2 - 12(3) + 18 = 2(9) - 36 + 18 = 18 - 36 + 18 = 0

["# Understanding the Quadratic Equation: f(3) = 2(3)² − 12(3) + 18 = 0", "When evaluating quadratic expressions, especially during test prep or algebraic practice, students often encounter expressions like f(3) = 2(3)² − 12(3) + 18 = 0. At first glance, this might seem like a straightforward calculation—but behind it lies a deeper exploration of function evaluation, quadratic functions, and verifying roots. In this article, we’ll unpack the calculation, explain the significance of the output (0), and highlight why this is a key concept in algebra and graphing quadratic functions.", "---", "## What is f(x)? A Quick Introduction to Quadratic Functions", "A quadratic function is typically written in the form:\n$$ f(x) = ax^2 + bx + c $$\nwhere ( a ), ( b ), and ( c ) are constants, and ( a <br/>\neq 0 ). The graph of a quadratic function is a parabola, which can open upward or downward depending on the sign of ( a ).", "Key features include:\n- Vertex: The maximum or minimum point of the parabola.\n- Zeros (or roots): Values of ( x ) where ( f(x) = 0 ), i.e., where the graph intersects the x-axis.", "---", "## Evaluating f(3): Step-by-Step Breakdown", "We begin with the function in point form:\nf(3) = 2(3)² − 12(3) + 18", "Let’s evaluate it step by step to confirm why it equals 0.", "1. Calculate the exponent:\n $$ (3)^2 = 9 $$\n So,\n $$ 2(9) = 18 $$", "2. Multiply:\n $$ -12(3) = -36 $$", "3. Combine all terms:\n $$ f(3) = 18 − 36 + 18 $$", "4. Perform arithmetic:\n $$ 18 − 36 = -18 $$\n Then,\n $$ -18 + 18 = 0 $$", "Thus, f(3) = 0, confirming that x = 3 is a root of the quadratic function.", "---", "## Why Is x = 3 a Root of the Function?", "Since f(3) = 0, we know that:\n$$ (x - 3) \ ext{ is a factor of the quadratic polynomial.} $$", "This means the quadratic has a factorization including ( (x - 3) ). To find the full expression, we can perform polynomial division or factorization. Let’s expand ( f(x) = 2x² − 12x + 18 ) and factor it.", "### Factor the quadratic expression", "Start with:\n$$ f(x) = 2x^2 - 12x + 18 $$", "Factor out the greatest common factor (GCF), which is 2:\n$$ f(x) = 2(x^2 - 6x + 9) $$", "Now, factor the trinomial:\n$$ x^2 - 6x + 9 = (x - 3)^2 $$", "So:\n$$ f(x) = 2(x - 3)^2 $$", "---", "## The Significance of f(3) = 0", "- x = 3 is a double root: Because it appears squared, the graph touches but does not cross the x-axis. The parabola is tangent to the x-axis at ( (3, 0) ), indicating a repeated root.\n- Symmetry and vertex: Since ( (x - 3)^2 ) is the factored form, the vertex occurs at ( x = 3 ), confirming that the vertex is exactly at this root.\n- Verification via Zero-Factor Theorem: The fact that ( f(3) = 0 ) confirms the factor theorem: if ( (x - 3) ) is a factor of ( f(x) ), then ( f(3) = 0 ), which we have verified.", "---", "## Real-World Implications and Graphical Behavior", "Understanding that ( f(3) = 0 ) helps in numerous applications:", "- Modeling phenomena: Quadratic equations model projectile motion, revenue projections, and area calculations. A root at ( x = 3 ) might indicate a critical point (e.g., time when height is zero in motion problems).\n- Symmetry: The repeated root implies symmetrical behavior around ( x = 3 ), useful for sketching graphs or predicting values.\n- Solving quadratics: Knowing all roots enables us to write the function’s full factored or expanded form and solve equations systematically.", "---", "## Conclusion: Mastering Quadratics Starts with Evaluation", "The expression f(3) = 2(3)² − 12(3) + 18 = 0 is more than a calculation—it’s a gateway to understanding quadratic behavior. By evaluating carefully and factoring the resulting expression, we uncover that x = 3 is a repeated root, a cornerstone concept in algebra. Whether for standardized tests, homework, or deeper math application, mastering function evaluation and root identification is essential.", "---", "## Key Takeaways", "- Always evaluate expressions step-by-step to avoid arithmetic errors.\n- A function value of zero indicates the input is a root or x-intercept.\n- Factoring reveals multiplicity of roots and symmetry in graphs.\n- ( f(3) = 0 ) confirms ( x = 3 ) is a solution and factor in the quadratic form.", "Master these steps, and you’ll confidently handle quadratic equations, interpret graphs, and apply algebra with clarity and precision.", "---", "Keywords for SEO: quadratic function evaluation, solve quadratic equation f(3)=0, verifying roots algebraically, factoring quadratics, repeated root meaning, function behavior graph, algebraic problem-solving."]