Problem 57 - Algebra 1: Concepts and Skills

Question

Evaluate the expression for the given value of the variable(s). $$x^{2}-12 when x=6$$

Step-by-Step Solution

Answer

The evaluated value of the expression is 24

Step 1: Substitute the given value

Replace the \(x\) in the equation \(x^{2} - 12\) with the value 6. This results in \(6^{2} - 12\)

Step 2: Calculate the squared term

Squaring the value 6 gives 36. The expression then becomes \(36 - 12\)

Step 3: Perform the subtraction

Subtract 12 from 36. The value obtained is 24

Key Concepts

Substitution

Substitution is a crucial concept in algebra, as it allows us to replace variables in an expression with specific values. In the given problem, we are asked to replace the variable \( x \) with \( 6 \). The expression originally is \( x^2 - 12 \). By substituting \( x \) with \( 6 \), the expression becomes \( 6^2 - 12 \). A few steps to remember during substitution:

• Identify the variable you need to substitute.
• Replace the variable with the given numerical value.
• Rewrite the complete expression with your substitution.

Substitution is often your first step in evaluating expressions, and mastering this can simplify more complex algebraic problems.

Evaluation

Evaluation involves finding the numerical value of an algebraic expression once substitution is complete. After substituting \( x \) with \( 6 \), the expression becomes \( 6^2 - 12 \). To evaluate:

• First, simplify each part of the substituted expression as much as possible.
• In our example, begin by squaring \( 6 \) to evaluate \( 6^2 \).
• Then proceed to perform any additional arithmetic operations.

Evaluation is a simple but detailed process, and practicing this helps in building a solid foundation in algebra. Being meticulous in stepping through these processes ensures greater accuracy in calculating expressions.

Squaring

The squaring operation is a fundamental aspect of algebra that involves multiplying a number by itself. In the expression \( 6^2 - 12 \), squaring is our first operation post-substitution. To square a number:

• Multiply the number by itself: for \( 6^2 \), compute \( 6 \times 6 \).
• This results in \( 36 \).

Understanding squaring is critical as it is common in many mathematical problems. Knowing how to quickly square numbers is helpful not just in algebra, but in many areas where exponential calculations are necessary.

Subtraction

Subtraction is the final arithmetic operation needed to evaluate our current expression. After we square \( 6 \) to get \( 36 \), we are left with the expression \( 36 - 12 \). To perform the subtraction:

• Take the result of the squared term and subtract \( 12 \) from it.
• In this case, compute \( 36 - 12 \), which equals \( 24 \).

Subtraction simplifies expressions by reducing numbers, and is essential in solving equations and simplifying calculations. Mastering subtraction ensures that you can accurately solve a wide range of mathematical problems.