Problem 39 - Intermediate Algebra

Question

Evaluate the algebraic expressions for the given values of the variables. $$ 2 a^{2}-a b+b^{2}, \quad a=-1 \text { and } b=-2 $$

Step-by-Step Solution

Answer

The expression evaluates to 4.

Step 1: Substitute the Variable Values

First, you substitute the given values of the variables into the algebraic expression. The expression is \(2a^2 - ab + b^2\). For \(a = -1\) and \(b = -2\), substitute these values:\[2(-1)^2 - (-1)(-2) + (-2)^2\]

Step 2: Evaluate the Powers

Now, calculate the powers for the variables that have been substituted. Calculate \((-1)^2\) which is 1, and \((-2)^2\) which is 4 .So the expression becomes:\[2(1) - (-1)(-2) + 4\]

Step 3: Perform Multiplications

Next, complete the multiplications in the expression.Multiply 2 by 1, which is 2, and calculate \(-1 \times -2\), which results in 2.Update the expression:\[2 - 2 + 4\]

Step 4: Simplify the Expression

Finally, simplify the expression by calculating the operations in the sequence:First, \(2 - 2\) results in 0.Then, \(0 + 4\) results in 4.So the expression evaluates to 4.

Key Concepts

Variable Substitution

Variable substitution is a fundamental concept when dealing with algebraic expressions. It involves replacing the variables in a mathematical expression with their given values. This allows the expression to be simplified into a numerical one. In our exercise, we have the expression \(2a^2 - ab + b^2\) and are given the values \(a = -1\) and \(b = -2\).

• The first step is to substitute \(a\) with \(-1\) and \(b\) with \(-2\) in the expression.
• This gives us the substituted expression: \(2(-1)^2 - (-1)(-2) + (-2)^2\).

Substituting variables correctly is crucial because it sets the stage for accurate evaluation. Without this step, you'd be working with abstract letters rather than concrete numbers.

Evaluating Expressions

Evaluating expressions means simplifying a substituted expression to find its numerical value. This involves following the order of operations known often by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

• After substituting the values in our expression, the next step is to evaluate any exponents. For \((-1)^2\), the result is 1, and for \((-2)^2\), it is 4.
• Our expression then becomes: \(2(1) - (-1)(-2) + 4\).

By systematically tackling the exponents first, you set yourself up for easier calculations in the later stages of evaluating the expression. It’s like solving a puzzle one piece at a time.

Mathematical Operations

Once the variables are substituted and any exponents are dealt with, it's time to perform the mathematical operations. This step involves multiplication, then addition and subtraction according to the order of operations:

• Multiply: \(2 \times 1\) gives you 2, and \(-1 \times -2\) gives you 2.
• After performing these multiplications, you're left with: \(2 - 2 + 4\).
• Finally, simplify the expression by performing addition and subtraction from left to right: \(2 - 2\) is 0, and then \(0 + 4\) results in 4.

Understanding how to correctly perform mathematical operations is key to finding the correct solution. Each operation builds on the one before it, guiding you to the final numerical value of the expression. By mastering these steps, you boost both your confidence and competence in algebra!