a_5 = 3(5)^2 - 2(5) + 1 = 3(25) - 10 + 1 = 75 - 10 + 1 = 66

Solving the Quadratic Expression: Step-by-Step Breakdown of 5², Parentheses, and Evaluation

Mathematics often feels like solving a puzzle, especially when working with algebraic expressions. One classic example is evaluating the expression:
5⁵ = 3(5)² - 2(5) + 1
But let’s correct and simplify this step-by-step for clarity, accuracy, and deeper understanding.

The Original Equation

The expression provided — a₅ = 3(5)² - 2(5) + 1 = 3(25) - 10 + 1 = 75 - 10 + 1 = 66 — begins with a mislabeled variable (a₅) but aims to evaluate a quadratic expression in terms of 5. While variable choice can vary, we’ll focus on computing:
3(5)² - 2(5) + 1

Understanding the Context

Step 1: Evaluate the Exponent (Order of Operations)

The first rule of exponents: parentheses and exponents come before multiplication. In the expression 3(5)², the (5)² means 5 squared.

5² = 25
So, 3(5)² = 3 × 25 = 75

Now the expression simplifies to:
75 - 2(5) + 1

a_5 = 3(5)^2 - 2(5) + 1 = 3(25) - 10 + 1 = 75 - 10 + 1 = 66

Key Insights

Step 2: Perform Multiplication

Next, multiply the remaining terms:

-2(5) = -10
Substitute back:
75 - 10 + 1

Step 3: Solve the Expression Left to Right

Now compute from left to right using basic arithmetic:

75 - 10 = 65
65 + 1 = 66

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Final Thoughts

Thus, 3(5)² - 2(5) + 1 = 66

Understanding the Mathematical Concept: Quadratic Expressions

The fully expanded expression, 3(5)² - 2(5) + 1, is a quadratic trinomial, a core concept in algebra. Quadratic expressions always take the form:
ax² + bx + c
Here, while no explicit x appears, we treat 5 as a placeholder variable, modeling how expressions depend on fixed values.

Evaluating such expressions helps students grasp:

Order of operations (PEMDAS/BODMAS)
Applying exponent rules
Simplifying complex algebraic expressions

The Role of Parentheses and Order of Operations

Parentheses group terms, ensuring correct computation order. In 3(5)², the exponent applies only to 5, not 3 — a crucial distinction. Failing to respect this leads to errors like 3(5)² = 3 × 5 × 2 = 30, which is wrong. Correctly, 5² first equals 25, multiplied by 3 gives 75.

Why This Expression Matters in Real Learning

Working through expressions like this builds foundational fluency critical for higher math:

Algebra: Prepare for solving equations and graphing functions.
Problem-solving: Train logical thinking under structured rules.
STEM readiness: Strengthens analytical skills used in physics, engineering, and computer science.

Conclusion

Evaluating 3(5)² - 2(5) + 1 isn’t just arithmetic — it’s about mastering expression manipulation, expectation alignment with order of operations, and laying groundwork for advanced algebra. Next time you see an expression labeled with variables, remember: follow the order, respect exponents, and simplify step by step — exactness leads to accuracy.