Problem 74 - Introductory Algebra for College Students

Question

Evaluate \(x^{2}-4\) for \(x=-3\)

Step-by-Step Solution

Answer

The value of the expression \(x^{2}-4\) when \(x=-3\) is \(5\).

Step 1: Substitute x with -3

Replace \(x\) in the expression \(x^{2}-4\) with \(-3\). This gives us \((-3)^{2}-4\).

Step 2: Square -3

Calculate the square of \(-3\) to update the expression. The square of \(-3\) is \((-3)*(-3)\), which equals \(9\). This changes our expression to \(9-4\).

Step 3: Subtract 4 from 9

Perform the subtraction operation between \(9\) and \(4\). This gives us \(9-4=5\).

Key Concepts

Substitution

Substitution is a fundamental concept in algebra that involves replacing a variable in an expression with a specific value. This process allows the expression to be evaluated numerically. For example, in evaluating the expression \(x^2 - 4\) for \(x = -3\), substitution helps us to find the result by inserting \(\(-3\)\) wherever \(x\) appears.
To substitute correctly, ensure that:

• You replace all instances of the variable with the given number.
• You handle negative numbers and parentheses accurately to avoid mistakes.

By effectively using substitution, you can transform algebraic expressions into simpler arithmetic problems that are easier to solve.

Squaring Numbers

Squaring a number means multiplying that number by itself. It is an essential operation in mathematics, commonly seen in polynomial expressions. For negative numbers, squaring always results in a positive value because multiplying two negative numbers yields a positive outcome.
Consider squaring \(-3\) in the expression \((-3)^2 - 4\):

• The square of \(-3\) is computed as \((-3)\times (-3)\).
• This calculation gives us \(9\), because the negative sign is negated when a negative number is multiplied by itself.

Understanding the squaring process is crucial for evaluating and simplifying expressions, especially when working with both negative and positive numbers.

Arithmetic Operations

Arithmetic operations are the basic building blocks of math calculations, including addition, subtraction, multiplication, and division. In evaluating expressions, these operations are used to simplify and find the value of the expression.
In our example, after squaring \(-3\), we get the expression \(9 - 4\). The next step involves simple subtraction:

• The operation \(9 - 4\) is carried out, which involves taking \(4\) away from \(9\).
• The result of this subtraction is \(5\).

Working through arithmetic operations step by step ensures accuracy in calculations. Mastery of these operations is vital for solving more complex mathematical problems.