Problem 56 - Algebra 1

Question

Use a calculator to evaluate the expression. Round your answer to two decimal places. $$4.7 b-\left(-b^{2}\right) \text { when } b=1.99$$

Step-by-Step Solution

Answer

The final answer to the equation is 13.31

Step 1: Substitute Value of b

The value of \( b \) given in the question is 1.99. Substitute \( b = 1.99 \) into the expression which becomes \( 4.7 \times 1.99 - (- 1.99^2) \).

Step 2: Solve Pararenthesis

Solve for \( - 1.99^2 \) in the equation giving \( 4.7 \times 1.99 - (- 3.96) \). Subtraction by a negative equates to addition.

Step 3: Perform Arithmetic Operations

Now perform the multiplication and addition operation, simplifying the equation to \( 9.353 + 3.96 \).

Step 4: Simplify to Decimal Places

Simplify to obtain the final answer. Ensure to round off to two decimal places as instructed.

Key Concepts

Expression Evaluation

Expression evaluation is the process of simplifying a mathematical statement to find its true value. In the world of algebra, expressions are combinations of numbers and variables connected by operations such as addition, subtraction, multiplication, and division.
When evaluating an expression, our goal is to find a single value that equals the expression when particular values are given for the variables. This often involves simplifying the expression through various mathematical operations.
In expressions like the one in this exercise, we need to understand each component and perform the calculations in a logical sequence. By doing so, we ensure the result is accurate and reflective of the given values. Keep in mind that knowing the rules of arithmetic operations and order can greatly influence the final outcome.

Substitution

Substitution is a critical concept in algebra that involves replacing a variable with a specific value. This process allows us to convert abstract expressions into concrete numbers, making them easier to handle.
For instance, in our exercise, the variable \( b \) is replaced with 1.99. This transformation changes the expression\( 4.7b - (-b^2)\) into \( 4.7 \times 1.99 - (-(1.99)^2)\).
Substitution is especially helpful when working with formulas and expressions where variables represent unknown or variable quantities. It's important to carefully substitute each occurrence of the variable in the expression to maintain its integrity. This accuracy ensures that when we move on to solving the expression, the results are precise.

Arithmetic Operations

Arithmetic operations are the foundational processes of calculating mathematical expressions. They include addition, subtraction, multiplication, and division. Each of these operations follows specific rules and order of precedence, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
In the given exercise, we perform a few crucial operations:

• Multiplication: Calculate \(4.7 \times 1.99\) to get part of the expression's value.
• Exponentiation: Compute \(- (1.99)^2 \) for the other part of the expression. Remember that subtracting a negative number is equivalent to adding it.

Next, sum the results of these operations to determine the expression's total value. Understanding arithmetic operations allows us to closely follow these steps and achieve a correct and accurate evaluation of expressions.

Rounding Numbers

Rounding numbers is a mathematical technique used to simplify figures by reducing the number of decimal places, providing a cleaner and often more comprehensible result. It's essential in situations where precision beyond a certain decimal point is unnecessary or constrained.
In our exercise, the final result needs to be expressed to two decimal places. This precision involves looking at the third decimal digit to decide whether to round up or down:

• If the third decimal digit is 5 or greater, increase the second decimal by one.
• If it’s less than five, retain the second decimal as is.

Rounding helps in communicating a precise yet concise value, which is crucial for accurate data interpretation and presentation, particularly in academic and professional settings.