Thus, $ p(x) = 2x^2 + x $. Then $ p(0) = 0 $.

["# Exploring the Quadratic Function ( p(x) = 2x^2 + x ): Evaluating ( p(0) = 0 )", "In the study of polynomial functions, quadratic equations play a foundational role due to their widespread applications in physics, engineering, economics, and mathematics. One such simple yet illustrative function is ( p(x) = 2x^2 + x ). This article explores the properties of this quadratic function, focusing particularly on the evaluation of ( p(0) ), which yields a clear and significant result: ( p(0) = 0 ).", "## Understanding ( p(x) = 2x^2 + x )", "The expression ( p(x) = 2x^2 + x ) defines a quadratic function where:", "- The coefficient of ( x^2 ) is ( 2 ) (a positive value indicating a parabola opening upwards),\n- The coefficient of ( x ) is ( 1 ),\n- There is no constant (free) term—the quadratic term starts at ( x^2 ).", "Quadratic functions generally take the standard form:", "[ p(x) = ax^2 + bx + c ]", "In this case, ( a = 2 ), ( b = 1 ), and ( c = 0 ). This structure allows us to analyze the behavior of the function, find its roots, compute its vertex, and evaluate specific values such as ( p(0) ).", "## Evaluating ( p(0) )", "To determine ( p(0) ), substitute ( x = 0 ) into the function:", "[\np(0) = 2(0)^2 + (0) = 0 + 0 = 0\n]", "Thus, ( p(0) = 0 ). This result is not only computationally simple but also conceptually meaningful. When the input is zero, both the linear and quadratic terms vanish, yielding zero for the entire expression. This zero value at ( x = 0 ) reflects the function’s origin in the coordinate plane—the point ( (0, 0) ) lies on the graph of ( p(x) ).", "Setting ( p(x) = 0 ) to find roots:", "[\n2x^2 + x = 0 \quad \Rightarrow \quad x(2x + 1) = 0\n]", "This gives two solutions: ( x = 0 ) and ( x = -\frac{1}{2} ). Indeed, ( p(0) = 0 ) confirms one of the roots, highlighting the function’s zero at the origin.", "## Why This Matters", "Evaluating ( p(0) ) in quadratic functions like ( p(x) = 2x^2 + x ) serves multiple purposes:", "- Initial Condition Analysis: In modeling real-world phenomena (e.g., projectile motion, revenue functions), knowing ( p(0) ) gives the starting point—here, zero output or zero initial value.\n- Graph Interpretation: The point ( (0, 0) ) is a root and often an intercept, shaping the shape and behavior of the parabola.\n- Numerical Verification: Confirming ( p(0) = 0 ) validates functional definitions and supports further algebraic or graphical exploration.", "## Conclusion", "The quadratic function ( p(x) = 2x^2 + x ) presents a clear example of how polynomial evaluation simplifies understanding function behavior. With ( p(0) = 0 ), we observe a zero at the origin—a critical insight into the function’s properties and applications. Whether for students learning algebra or professionals applying math models, recognizing such fundamental evaluations strengthens mathematical fluency and problem-solving clarity.", "---", "Keywords: ( p(x) = 2x^2 + x ), quadratic function, evaluate ( p(0) ), zero of function, polynomial evaluation, algebra, coordinate geometry."]