Problem 45 - Fundamentals of Mathematics

Question

For problems \(17-46\), find the value of each expression. $$ (y+2)^{2}-6(y+2)-6, \text { if } y=2 $$

Step-by-Step Solution

Answer

The value of the expression is -14.

Step 1: Substitute the given value into the expression

The expression we have is \((y+2)^2 - 6(y+2) - 6\). We are given \(y = 2\). Substitute 2 into the expression to get: \((2+2)^2 - 6(2+2) - 6\).

Step 2: Simplify inside the parentheses

First, calculate \(2 + 2 = 4\). The expression now becomes \(4^2 - 6 \times 4 - 6\).

Step 3: Calculate the square and multiplication

Calculate \(4^2\), which is 16, and \(6 \times 4\), which is 24. The expression is now \(16 - 24 - 6\).

Step 4: Perform the subtraction

Now, subtract 24 from 16 to get \(-8\). Then subtract 6 to get \(-8 - 6 = -14\).

Key Concepts

Substitution

Substitution is a fundamental concept in algebra that involves replacing variables with specific values. This technique is essential when evaluating expressions or equations, allowing us to find numerical values when certain conditions are given. In our exercise, we applied substitution by replacing the variable \(y\) in the expression \((y+2)^2 - 6(y+2) - 6\) with the value 2. This was done because the problem provided \(y = 2\). To substitute correctly:

• Identify the variable to be replaced.
• Take the given value (in this case, 2) and substitute it into every occurrence of the variable within the expression.

After performing the substitution, the expression transformed into \((2+2)^2 - 6(2+2) - 6\). This step is crucial because it simplifies the process into a purely numerical calculation, paving the path to find the expression's value.

Simplifying Expressions

Simplifying expressions is a key skill in algebra that helps make complex expressions more manageable and easier to evaluate. In essence, simplifying involves performing arithmetic operations within the expression to reduce it to its most basic form. In the exercise example, after substitution, we had the expression \((2+2)^2 - 6(2+2) - 6\). To simplify, follow these steps:

• First, perform any operations inside parentheses. Here, calculate \(2+2\), resulting in 4.
• Next, apply arithmetic operations like exponentiation and multiplication. Square the 4 to get 16 and multiply 6 by 4 to get 24.

Hence, the expression was simplified from \((2+2)^2 - 6(2+2) - 6\) to \(16 - 24 - 6\). This reduction to simpler terms allows for straightforward arithmetic calculations to arrive at the final answer.

Order of Operations

The order of operations is a fundamental principle in mathematics that dictates the sequence in which calculations should be performed to ensure consistent and correct results. Often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), it guides us on how to approach calculations systematically.In our solved exercise, the following steps were performed according to the order of operations:

• First, handle operations within Parentheses: \((2+2)\).
• Next, calculate Exponents: \(4^2\) giving 16.
• Then, perform Multiplication: \(6 \times 4\) resulting in 24.
• Finally, conduct Subtraction: sequentially calculate \(16 - 24 - 6\).

By adhering to the order of operations, the expression was methodically reduced to a single final value of \(-14\). This systematic approach prevents errors and maintains the integrity of the calculations.