Problem 73 - Algebra 1

Question

Evaluate the expression to the nearest hundredth. $$ \frac{-3 \pm 2 \sqrt{5}}{-1} $$

Step-by-Step Solution

Answer

The expression \(\frac{-3 \pm 2 \sqrt{5}}{-1}\) evaluates to -1.48 and 7.48 to the nearest hundredth.

Step 1: Evaluate the square root

The first step is to evaluate the square root \(\sqrt{5}\), which equals approximately 2.24.

Step 2: Apply the plus-minus operation

After getting the square root result, you need to apply the plus-minus operation into the equation, which would give us two parts that need to be separately solved: \(\frac{-3 + 2 * 2.24}{-1}\) and \(\frac{-3 - 2 * 2.24}{-1}\).

Step 3: Perform the multiplication

Next, perform the multiplication operation in both equations. This gives us \(\frac{-3 + 4.48}{-1}\) and \(\frac{-3 - 4.48}{-1}\).

Step 4: Perform the addition and subtraction

Next step is to perform the addition in the first equation and subtraction in the second equation, resulting in the equations: \(\frac{1.48}{-1}\) and \(\frac{-7.48}{-1}\).

Step 5: Perform the division

The final step is to complete the division operation, which yields -1.48 and 7.48 respectively.

Key Concepts

Understanding the Plus-Minus Operation

The plus-minus operation, denoted as \( \pm \), is a critical concept in solving quadratic equations and expressions involving radicals. It indicates that an expression can take on two different values: one where the terms are added, and another where they are subtracted. In the context of our exercise, this operation requires you to solve the expression twice, once with the plus (\( + \)) and once with the minus (\( - \)).
For example, in the expression \( \frac{-3 \pm 2\sqrt{5}}{-1} \), the plus-minus operation results in two expressions to solve: \( \frac{-3 + 2\sqrt{5}}{-1} \) and \( \frac{-3 - 2\sqrt{5}}{-1} \).
This dual solution approach is essential, particularly when working with quadratic equations, as it often leads to two distinct answers. Remember, applying the plus-minus operation effectively doubles the required calculations but is crucial for achieving comprehensive solutions.

Demystifying Square Root Calculation

Square root calculation is a fundamental part of solving quadratic equations, especially when working with radicals. To calculate the square root, you’re identifying a number that, when multiplied by itself, gives the original number. In our example, finding \( \sqrt{5} \) is the first step.
Using a calculator, the square root of 5 is approximately equal to 2.24. This decimal approximation helps you advance through further computational steps.
Accuracy is key in square root calculations to ensure subsequent operations, such as multiplication and addition or subtraction, are correct in your math journey. While recognizing the approximate value is often enough when solving these types of problems, maintaining precision ensures the end result is as accurate as possible.

Clarifying the Division Operation

The division operation in algebra involves splitting a number into equal parts, called the divisor. In algebraic expressions, division simplifies ratios or fractions, like in the exercise expression \( \frac{-3 \pm 2\sqrt{5}}{-1} \).
Applying the division operation means you divide each result from previous calculations by \(-1\). The division by a negative flips the sign of the quotient. For instance:

• The equation \( \frac{1.48}{-1} \) becomes \(-1.48\) when divided.
• Similarly, \( \frac{-7.48}{-1} \) results in \( 7.48 \).

This step is crucial as it determines the final values of the solution, showing how each operation intricately connects to form the complete solution.

Unpacking Addition and Subtraction Operations

Addition and subtraction operations are the foundation of many algebraic processes. They involve bringing quantities together or taking them apart, respectively. In this scenario, after multiplying, you perform these operations to simplify the terms:
From multiplication, we obtained \(-3 + 4.48\) and \(-3 - 4.48\).
Breaking it down:

• For \(-3 + 4.48\), you add the two values, resulting in \(1.48\).
• For \(-3 - 4.48\), you subtract, leading to \(-7.48\).

The precision in executing these operations directly affects the results. These steps smoothly transition the problem from logical setup to a form where you can perform the division operation, leading to the final answer in solving the expression.