Problem 75 - Introductory Algebra for College Students

Question

In Exercises \(73-80,\) evaluate each algebraic expression for the given value of the variable. $$3 x^{2}-8 x ; x=-2$$

Step-by-Step Solution

Answer

The evaluation of the expression \(3 x^{2}-8 x\) for \(x = -2\) is 28

Step 1: Substitute the given value into the algebraic expression

Substitute the value of the variable \(x = -2\) into the given algebraic expression \(3 x^{2}-8 x\). This yields \(3 (-2)^{2}-8 (-2)\)

Step 2: Apply the order of operations

According to the rules of the order of operations, exponential operations should be carried out first. So the square of -2 is calculated resulting in \(3×4 + (8×2) = 12 + 16\)

Step 3: Simplify the expression to get the final result

By adding up the previous result \(12 + 16\) we obtain the final result, which is 28. This is the evaluation of the expression for \(x = -2\).

Key Concepts

Understanding Order of Operations

When working with algebraic expressions, it's crucial to follow the order of operations to ensure accurate calculations. This sequence is often remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). Let's break it down more simply:

• Parentheses: Always start by solving expressions inside parentheses first.
• Exponents: Calculate powers and square roots before anything else.
• Multiplication and Division: Perform these operations next, moving from left to right across the expression.
• Addition and Subtraction: Finally, solve any addition or subtraction tasks, moving from left to right.

In our exercise, after substituting the value into the expression, we calculated the exponent next because powers are higher in the order of operations. This ensures that calculations are completed in the correct sequence, preventing potential errors in the results.

Mastering Substitution

Substitution is a fundamental skill in algebra where you replace a variable with a given value. It simplifies expressions and helps in evaluating them. In our example, we began by substituting the given value of the variable, which was \(x = -2\), into the algebraic expression \(3x^2 - 8x\).Once the substitution was made, the expression \(3(-2)^2 - 8(-2)\) became ready for further simplification. Here are some tips to consider during substitution:

• Write Clearly: After substitution, ensure the expression is clearly written out to avoid mistakes.
• Check Your Work: Always recheck the substituted values and the modified expression for accuracy.
• Use Parentheses: Especially when substituting negative numbers, use parentheses to keep the expression accurate and neat.

Substitution allows us to convert the expression into something manageable and straightforward for further calculations.

Evaluating Expressions

Evaluating an algebraic expression involves substituting the variables with their values and then simplifying the expression using the order of operations. This process converts an algebraic phrase into a single numerical value. Let's consider the steps taken in our exercise:Substitute \(x = -2\) into the expression \(3x^2 - 8x\). You'll have \(3(-2)^2 - 8(-2)\).Follow the order of operations:

• First, calculate the exponent \((-2)^2 = 4\), making the expression \(3 \times 4 - 8(-2)\).
• Next, perform multiplication: \(3 \times 4 = 12\) and \(-8 \times -2 = 16\).
• Finally, add these results together: \(12 + 16 = 28\).

By following these steps, we arrived at the value \(28\) for the expression when \(x = -2\). Evaluating expressions requires attention to detail and methodical application of mathematical principles to achieve accurate results.