Problem 72 - Algebra 1

Question

Evaluate the expression. $$t^{2}-7 t+12 \text { when } t=8$$

Step-by-Step Solution

Answer

The final evaluation of the expression when \(t=8\) is 20.

Step 1: Substitution

Replace \(t\) with 8 in the expression. This gives: \(8^{2} - 7*8 + 12\).

Step 2: Follow Operational Order

Perform the operations in the correct order. Start with the exponent, then perform multiplication and at last addition. This results in: \(64 - 56 + 12\).

Step 3: Perform Addition/Subtraction

Continue performing operations, doing addition and subtraction from left to right. This results in: \((64 - 56) + 12 = 8 + 12 = 20\).

Key Concepts

Substitution

In mathematics, substitution is a straightforward concept. You simply replace a variable in an expression with a given value. In this exercise, the variable is \( t \), and we substitute it with the number 8. This transforms the original expression \( t^2 - 7t + 12 \) into \( 8^2 - 7 \times 8 + 12 \).

Substitution is like putting a specific number into a recipe where a variable is used. It's important to make sure every occurrence of the variable is replaced with the number you are given.

For this concept, make sure to double-check the transformation of each term to ensure accuracy. A small mistake can lead to an incorrect solution.

Order of Operations

The order of operations is a critical rule in mathematics. It tells us which calculations to perform first in an expression to ensure everyone gets the same answer. In this exercise, after substituting \( t = 8 \), the expression becomes \( 8^2 - 7 \times 8 + 12 \).

The order to follow is often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)). Here’s the process for our expression:

• Exponents: Calculate \( 8^2 \) which is 64.
• Multiplication: Compute \( 7 \times 8 \), resulting in 56.
• Addition/Subtraction: Perform from left to right: \( 64 - 56 + 12 \).

This correct order ensures accuracy and consistency.

Polynomial Evaluation

Evaluating a polynomial is the process of calculating the output of the polynomial given a particular input for its variable. The exercise in question involves a polynomial expression \( t^2 - 7t + 12 \) with \( t = 8 \).

Once the value (8) is substituted into the expression, following the order of operations as described allows us to efficiently simplify and compute:

• The expression converts to \( 8^2 - 7 \times 8 + 12 \).
• Using the order of operations: \( 64 - 56 + 12 \).
• This breaks down further into \( 8 + 12 = 20 \).

Polynomial evaluation involves careful substitution and following of operations steps. Once mastered, it becomes a valuable skill in algebra and beyond.