Problem 26 - Algebra 2

Question

Evaluate each expression if \(a=\frac{2}{5}, b=-3, c=0.5,\) and \(d=6\). \((5-d)^{2}+a\)

Step-by-Step Solution

Answer

The expression evaluates to \(\frac{7}{5}\).

Step 1: Evaluate the Expression Inside the Parentheses

The expression inside the parentheses is \((5-d)\). Substitute \(d = 6\) into the expression: \(5 - 6 = -1\). Thus, \((5-d) = -1\).

Step 2: Calculate the Square of the Result

Now that we have \((5-d) = -1\), calculate \((-1)^2\). The square of \(-1\) is 1, so \((-1)^2 = 1\).

Step 3: Add the Value of a

Now add \(a\) to the result of the previous step. Given \(a = \frac{2}{5}\), calculate \(1 + \frac{2}{5}\). Convert the whole number 1 to a fraction \(\frac{5}{5}\) to keep a common denominator: \(\frac{5}{5} + \frac{2}{5} = \frac{7}{5}\).

Step 4: Final Result

Combining all steps, we find the expression evaluates to \(\frac{7}{5}\).

Key Concepts

Substitution

Substitution is a core concept in algebra that involves replacing variables in an expression with their given values. This step is crucial in simplifying expressions and solving equations. Let's take a closer look at this process using our example:

- We have the expression \((5-d)^{2}+a\), and we need to substitute the provided value for \(d\), which is \(6\).- By substituting \(d = 6\), we transform the expression inside the parentheses from \((5-d)\) to \((5-6)\), which simplifies to \(-1\).
This substitution step sets the stage for further simplifications and is a vital skill when dealing with more complex expressions.

Using substitution effectively can help you understand how different variables affect an expression's result and is an essential foundational skill in algebra.

Order of Operations

The order of operations is a set of rules that dictates the correct sequence to follow when evaluating expressions. It helps ensure that everyone arrives at the same result when performing calculations. The standard order of operations can be remembered using the acronym PEMDAS:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

In our exercise, we first handle the parentheses: \(5-d\) was evaluated to \(-1\). Following that, we had to deal with the exponent, which involves squaring the result:

- The square of \(-1\) is calculated as \((-1)^{2}=1\).- Finally, we add \(a\) to the squared value, ensuring all operations are performed in the correct sequence.
Adhering to the order of operations is essential for obtaining the correct results in mathematical expressions.

Squaring Numbers

Squaring a number means multiplying the number by itself. This operation is a type of exponentiation where the exponent is 2. Squaring is commonly used in various mathematical and scientific calculations.

In our example, after substituting our variables into the expression, we had to square the result of \(5-d\), which was \(-1\). Squaring \(-1\) involves:

• Calculating \((-1) \times (-1) = 1\)

It's important to remember that squaring a negative number results in a positive value. This is because the two negative factors cancel each other out, a key concept that applies to all real numbers. Recognizing this property helps avoid common mistakes when working with expressions that contain negative terms squared.
Understanding squaring and its implications are fundamental for tackling complex algebraic problems.