Problem 83 - Algebra 1: Concepts and Skills

Question

Use a calculator to evaluate the expression. Round the results to the nearest hundredth. $$ \frac{7 \pm 3 \sqrt{12}}{-6} $$

Step-by-Step Solution

Answer

The two possible answers, rounded to the nearest hundredth, should be entered here. Please check your calculator for the results.

Step 1: Understand The Symbols And Evaluate The Expression With Positive Sign

First, it's important to recognize the plus-minus symbol (±). This means there will be two results - one with addition and one with subtraction. Let's start with the positive sign. Using a calculator, substitute \(+3\) for \(±\) in the expression and perform the operation: \(\frac{7 + 3 \sqrt{12}}{-6}\).

Step 2: Round The Result

After performing the operation, you should have a decimal number. Round this to the nearest hundredth - that is, the second decimal place.

Step 3: Evaluate The Expression With Negative Sign

Repeat the same procedure but this time using the negative sign. So, substitute \(-3\) for \(±\) in the expression and perform the operation: \(\frac{7 - 3 \sqrt{12}}{-6}\).

Step 4: Round The Second Result

Finally, round your second result to the nearest hundredth.

Key Concepts

Order of Operations

Understanding the order of operations is essential in mathematics, especially when handling complex expressions. The order of operations determines the sequence in which components of a math problem are tackled to ensure a correct answer. This sequence is often remembered by the acronym PEMDAS:

• P: Parentheses - Solve expressions within parentheses first.
• E: Exponents - Next, handle any exponents or powers.
• M and D: Multiplication and Division - Perform these from left to right.
• A and S: Addition and Subtraction - Lastly, tackle addition and subtraction from left to right.

In the given expression \(\frac{7 \pm 3 \sqrt{12}}{-6}\), the operations are tackled in this order: evaluate the square root, then any multiplication or division, and finally, handle the addition or subtraction. Following these steps ensures that the calculations are accurate and consistent.

Rounding Decimals

Rounding decimals is a common task in mathematics to simplify numbers or fit them within certain constraints. Rounding to the nearest hundredth means we focus on the second digit to the right of the decimal point. Here's a simple guide on how to round decimals:

• Identify the digit in the desired place (for hundredths, this is the second decimal place).
• Look at the digit immediately to the right (the thousandths place).
• If this digit is 5 or greater, add 1 to the hundredths place. If it's less than 5, leave the hundredths digit unchanged.

So, when given a number like 2.456 and asked to round to the nearest hundredth, observe the thousandths place (6). Since 6 is greater than 5, you round up to 2.46.

Square Roots

Square roots are fundamental in math, often used to explore relationships between numbers and simplify expressions. The square root of a number \(a\) is a value that, when multiplied by itself, equals \(a\). For instance, the square root of 9 is 3 because \(3 \times 3 = 9\). We express square roots with the radical sign, \(\sqrt{}\).To find a square root such as \(\sqrt{12}\), follow these steps:

• Approximate: Since 9 is the nearest perfect square below 12, and 16 is the next above it, \(\sqrt{12}\) is between \(\sqrt{9}=3\) and \(\sqrt{16}=4\).
• Use a calculator for an accurate decimal value: \(\sqrt{12} \approx 3.4641\).

Understanding square roots allows us to solve the expression \(\frac{7 \pm 3 \sqrt{12}}{-6}\) accurately, by calculating \(3 \times \sqrt{12}\) first.

Plus-Minus Symbol

The plus-minus symbol (±) indicates two possible operations in an expression, representing both addition and subtraction. It's a way to express a pair of solutions or results at once. In our expression \(\frac{7 \pm 3 \sqrt{12}}{-6}\), this symbol signifies that we need to calculate two separate expressions:

• One using the plus sign: \(\frac{7 + 3 \sqrt{12}}{-6}\)
• And one using the minus sign: \(\frac{7 - 3 \sqrt{12}}{-6}\)

Understanding the plus-minus symbol helps you solve problems that have multiple outcomes, catering to both scenarios in calculations. This approach expands your ability to interpret and calculate varied results in mathematical expressions.