Problem 32 - Elementary Algebra

Question

Evaluate. \(a_{2}-b_{2},\) where \(a=-1\) and \(b=-2\)

Step-by-Step Solution

Answer

The expression evaluates to -3.

Step 1: Substitute the Values

We start by substituting the given values of \(a\) and \(b\) into the expression \(a_2 - b_2\). Since we want \(a^2\) and \(b^2\), substitute \(a = -1\) and \(b = -2\).

Step 2: Calculate \(a^2\)

Compute the square of \(a\) as \((-1)^2 = 1\). This gives us the value of \(a^2 = 1\).

Step 3: Calculate \(b^2\)

Compute the square of \(b\) as \((-2)^2 = 4\). This means \(b^2 = 4\).

Step 4: Evaluate the Expression

Now that we have the values of \(a^2\) and \(b^2\), substitute them back into the expression: \(a^2 - b^2 = 1 - 4\).

Step 5: Simplify the Result

Subtract the values: \(1 - 4 = -3\). The solution to the expression is \(-3\).

Key Concepts

Evaluation

Evaluation is a critical concept in algebra, involving finding the value of an expression by following a series of logical steps. The process helps us understand how expressions change when variables are replaced with specific values. In the exercise, we evaluate the expression \(a_{2} - b_{2}\) by using the given values of \(a\) and \(b\). Let's break it down:

• Start by replacing each variable in the expression with their assigned values. In this case, substitute \(a = -1\) and \(b = -2\).
• This step is vital because it turns an algebraic expression with variables into a numeric one, making it solvable.
• After substituting the values, you can proceed to other arithmetic operations like addition, subtraction, multiplication, or division to get the final answer.

By evaluating expressions, we gain a deeper understanding of how different values impact the outcomes of algebraic operations. This foundational skill is crucial for tackling more complex mathematical problems later on.

Substitution Method

The substitution method is an efficient technique used in algebra to simplify or solve expressions and equations. In this method, values are inserted in place of variables, which then allows you to perform computations step-by-step to arrive at a solution. This method holds particular importance when dealing with expressions that involve variables being squared or any other operation.

• First, identify the variables in your expression. In the given exercise, these are \(a\) and \(b\).
• Next, replace these variables with their corresponding values as provided. Substitute \(a\) with \(-1\) and \(b\) with \(-2\).
• Finally, compute the expression with positive numbers. This substitution simplifies calculations and helps to clearly see the progression from an algebraic to a numerical answer.

Using the substitution method is a simple yet powerful way to clarify and solve expressions. It reduces the complexity of algebraic operations, making them more approachable and easier to handle.

Squared Terms

Squared terms are a vital component in algebraic expressions, particularly those involving powers of numbers. A term is squared when it is multiplied by itself. The squaring operation has unique properties that significantly affect calculations:

• The square of a negative number results in a positive value. For instance, \((-1)^2 = 1\) and \((-2)^2 = 4\), as the negative signs cancel out when multiplied.
• Squaring is an example of an exponent operation; here, the exponent is 2. It involves raising the base number to a power, exemplified by \(a^2\) and \(b^2\) in our exercise.
• Understanding squared terms is essential because they frequently appear in equations that describe real-world phenomena, from physics to finance.

Grasping the concept of squared terms allows students to successfully navigate through various algebraic problems and develop confidence in handling polynomial expressions.