Understanding the Quadratic Expression: S(3) = 2(3)² + 3(3) + 1 = 28

When solving quadratic expressions, one common form students and math enthusiasts encounter is S(n) = an² + bn + c. A recent example that illustrates this pattern clearly is S(3) = 2(3)² + 3(3) + 1, which simplifies step-by-step to yield the final result of 28. Let’s break down this calculation and explore what makes this quadratic expression meaningful.

Understanding the Context

Breaking Down the Expression: S(3) = 2(3)² + 3(3) + 1

This quadratic function fits the standard form:
S(n) = an² + bn + c
where:

  • a = 2
  • b = 3
  • c = 1

When evaluating S(3), we substitute n = 3 into the expression:
S(3) = 2(3)² + 3(3) + 1

Step-by-step evaluation:

S(3) = 2(3)^2 + 3(3) + 1 = 2(9) + 9 + 1 = 18 + 9 + 1 = 28

Key Insights

  1. Calculate the squared term:
    (3)² = 9
    Then multiply by coefficient a:
    2 × 9 = 18

  2. Evaluate the linear term:
    3 × 3 = 9

  3. Add all parts:
    18 + 9 + 1 = 28

Thus, S(3) = 28, confirming the complete simplification.

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Final Thoughts

Why This Example Matters

This particular quadratic showcases key mathematical concepts:

  • Polynomial evaluation: Substituting a value into a quadratic to find its output.
  • Order of operations: Proper handling of exponents, multiplication, and addition.
  • Real-world applications: Such expressions model several phenomena in physics, engineering, and economics, like projectile motion or profit calculations.

Understanding how to compute entities like S(3) helps build foundational algebra skills, crucial for advanced math topics such as calculus, optimization, and data analysis.

Conclusion

The expression S(3) = 2(3)² + 3(3) + 1 = 28 elegantly demonstrates the power and clarity of quadratic functions. By breaking down each term, learners can reinforce their algebraic fluency and prepare for more complex mathematical challenges.

Whether you're a student mastering algebra or a teacher seeking clear examples, breaking expressions into step-by-step components demystifies quadratics and highlights their practical value.

Learn more about quadratic functions and algebraic simplification in our comprehensive guides — unlocking clarity in every step!