Problem 38 - Algebra for College Students

Question

Evaluate the algebraic expressions in Problems 35-57 for the given values of the variables. \(3 a^{2}+2 b^{2}, \quad a=2\) and \(b=5\)

Step-by-Step Solution

Answer

The value of the expression is 62.

Step 1: Identify and Substitute

First, identify the algebraic expression to be evaluated, which is \(3a^2 + 2b^2\). Then, substitute the given values, \(a = 2\) and \(b = 5\), into the expression.

Step 2: Perform Substitution

Substituting the given values into the expression, we have \(3(2)^2 + 2(5)^2\). This means we need to replace \(a\) with 2 and \(b\) with 5.

Step 3: Calculate Squared Terms

Calculate the squared terms: \(2^2 = 4\) and \(5^2 = 25\). Substitute these results back into the expression, leading to \(3 \times 4 + 2 \times 25\).

Step 4: Perform Multiplications

Multiply the coefficients by the squared terms: \(3 \times 4 = 12\) and \(2 \times 25 = 50\).

Step 5: Add Results

Add the results of the multiplications: \(12 + 50\). This gives 62.

Key Concepts

Variable Substitution

Variable substitution is the first and crucial step when evaluating algebraic expressions. It involves replacing variables with given numbers. In the exercise, you had to evaluate the expression \(3a^2 + 2b^2\) using \(a = 2\) and \(b = 5\). Here's how you do it:

• Identify the variables in the expression. In this case, they are \(a\) and \(b\).
• Once identified, substitute each occurrence of \(a\) with 2 and each occurrence of \(b\) with 5.

After substitution, your expression changes from \(3a^2 + 2b^2\) to \(3(2)^2 + 2(5)^2\). This step sets the stage for further calculations by making the expression numerical, thereby eliminating the variables.

Squared Terms

After substituting the variables, the next step is to calculate the squared terms. Squaring a number means multiplying the number by itself. This is an essential step in dealing with algebraic expressions that contain squared variables. Here's the process:

• For \(a^2\), it means \(2^2 = 2 \times 2 = 4\).
• For \(b^2\), it means \(5^2 = 5 \times 5 = 25\).

Once you've calculated these squared terms, substitute back into your expression. Your expression now becomes a simpler numeric expression: \(3 \times 4 + 2 \times 25\). This transformation is essential as it simplifies further calculations.

Algebraic Operations

The final stage involves performing the basic arithmetic operations, which in this case are multiplication and addition. This is where you actually evaluate the expression into a final numeric result.Here's how to proceed:

• First, handle the multiplications. Use the coefficients present in the expression: \(3 \times 4\) and \(2 \times 25\).
• This gives two separate products: 12 and 50.
• Finally, add these products together: \(12 + 50 = 62\).

The result, 62, is the value of the original algebraic expression with the provided values of the variables. Understanding the flow of operations helps you gain confidence in evaluating similar expressions effortlessly.