Problem 55 - Algebra 1: Concepts and Skills

Question

Evaluate the expression \(32-x^{2}+9\) when \(x=2.\) $$F)19\quad G)21\quad H)37\quad J)39$$

Step-by-Step Solution

Answer

The value of the expression when x is 2 is 37. So, the correct answer is H)37

Step 1: Substitute x into the equation

Replace the variable x in the equation \(32-x^{2}+9\) with 2. This creates the equation \(32-2^{2}+9\).

Step 2: Solve the exponent first

According to the order of operations (PEMDAS/BODMAS), solve the exponent before performing the subtraction. \(2^{2}=4\), so now the equation becomes \(32-4+9\).

Step 3: Perform the subtraction and addition

Subtract 4 from 32 to get 28 and then add 9 to 28. The expression equals to 37.

Key Concepts

Order of Operations

Understanding the order of operations is crucial when evaluating algebraic expressions. It ensures that calculations are performed in a consistent manner and prevents ambiguity in mathematical communication. The common acronym PEMDAS is used in the United States, standing for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). An alternative acronym is BODMAS, which stands for Brackets, Orders (i.e., Powers and Square Roots, etc.), Division and Multiplication (left to right), Addition and Subtraction (left to right).

Always start with operations inside parentheses or brackets, then move on to exponents, before proceeding to multiplication and division, and finally address addition and subtraction. This hierarchy controls the procedural steps in evaluating an expression like \(32-x^{2}+9\) and helps avoid common mistakes that can lead to incorrect answers.

Substituting Variables

Substituting variables into algebraic expressions is a method to find the value of expressions for certain numerical values of the variables. This is like replacing placeholders with concrete numbers. Here are some essential steps to follow:

• Identify the variable(s) in the expression.
• Determine the value of each variable from the given information.
• Replace the variable in the expression with its corresponding value.
• Ensure that you substitute the value everywhere the variable appears in the expression.

For example, when asked to evaluate \(32-x^{2}+9\) with \(x=2\), the substitution stage involves replacing 'x' with 2. This is a critical step for proceeding with accurate calculations in the expression.

Solving Exponents

Exponents denote repeated multiplication and are crucial in simplifying expressions before performing other operations. The expression \(a^n\) means 'multiply a by itself n times'. When solving exponents, remember the following:

• The exponent tells you how many times to use the base as a multiplier.
• If an exponent is positive, multiply the base number by itself as many times as the exponent specifies.
• Dealing with exponents is typically done after parentheses but before multiplication, division, addition, and subtraction in the order of operations.

In our example, the exponent is solved as \(2^{2} = 2 \times 2 = 4\), reducing the expression to \(32-4+9\), ready for the next operations.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x or y), and operators (like add, subtract, multiply, and divide). These expressions can range from simple to very complex. It's important to recognize that unlike equations, expressions do not have a 'equals' sign and are not 'solved' but 'simplified' or 'evaluated'.

To successfully evaluate an expression, all variables must be substituted with their respective values, if known, and then simplified using the order of operations. In our exercise, after substituting and solving for the exponent, we simplify the resulting arithmetic expression to find the value of the original algebraic expression.