Problem 69 - A Graphical Approach to College Algebra

Question

Approximate each expression to the nearest hundredth. $$3(5.9)^{2}-2(5.9)+6$$

Step-by-Step Solution

Answer

The expression approximates to 98.63.

Step 1: Square the number inside the expression

First, calculate the square of the number inside the expression, which is \( (5.9)^2 \). Use a calculator to find \( 5.9 \times 5.9 = 34.81 \).

Step 2: Multiply by the coefficient

Next, multiply the result of the squaring by the coefficient 3. Calculate \( 3 imes 34.81 = 104.43 \).

Step 3: Multiply for the linear term

Calculate the multiplication of the linear term by its coefficient, \( 2(5.9) \). Do \( 2 imes 5.9 = 11.8 \).

Step 4: Substitute and simplify the expression

Substitute the results from the previous steps into the original equation: \( 3(5.9)^2 - 2(5.9) + 6 \) becomes \( 104.43 - 11.8 + 6 \). Simplify this by performing the operations: \( 104.43 - 11.8 = 92.63 \), and then \( 92.63 + 6 = 98.63 \).

Step 5: Round to the nearest hundredth

Finally, round the result \( 98.63 \) to the nearest hundredth. As it's already at two decimal places with no need for further rounding, the answer remains \( 98.63 \).

Key Concepts

Step by Step Solution

A step-by-step solution gives you a clear path to follow when solving algebraic expressions, like the one in our exercise. It's similar to following a recipe where each instruction builds on the previous one. By understanding each part of the calculation process, you can solve even complex mathematical problems.

• Start by identifying parts of the expression. This involves recognizing terms, coefficients, and operations.
• Perform calculations sequentially. It's important to follow the order of operations, also known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
• Substitute back into the equation. After resolving individual tasks, combine them to simplify the entire expression.
• Check your work for accuracy. Ensure all calculations are correct, and confirm if rounding is needed as in the final step of this specific problem.

Following a step-by-step approach can demystify algebraic expressions and help you secure accurate solutions. Taking the process one step at a time makes complex problems manageable.

Approximation

Approximation is a mathematical technique used to find a number close to the exact mathematical value but easier to use. In problems like ours, approximation helps us get a user-friendly result quickly. Here’s how you should think about it:

• Understand decimal places. Numbers might have a lot of decimal places, and not all are necessary for every problem.
• Rounding rules. When a task requires rounding to, for example, the nearest hundredth, you only look at two decimal places. If the third decimal digit is 5 or greater, you increase the second one by four. If it is less than 5, keep the original two-digit value.

Using approximation isn't just about making life easier—it's essential when precision beyond a certain point doesn't impact the result significantly. This is commonly used in real-life scenarios, like when you're dealing with money, measurements, and more.

Mathematical Operations

Mathematical operations are the building blocks of algebra and are crucial to solving expressions like the one in the exercise. They involve basic yet foundational functions in mathematics.

• Addition and Subtraction: These operations help combine or separate values. They are fundamental in simplifying expressions to reach a final answer.
• Multiplication and Division: Used to scale numbers up or down. Recognizing when to use these operations is key in manipulating terms efficiently.
• Exponents: Essential for squaring numbers. An exponent, like the one seen in the expression (5.9)^2, means multiplying the number by itself.

Dissecting complex expressions into these mathematical operations helps in understanding the overall structure and flow of a problem. Each operation plays a specific role that when pieced together, leads to solving the problem effectively.