Solving The Equation Sqrt(-4x) = 100 A Step By Step Guide

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Introduction

In this article, we will delve into the process of solving the equation ${ \sqrt{-4x} = 100 }$. This mathematical problem falls under the category of algebra and involves dealing with square roots and negative numbers. Understanding how to solve such equations is crucial for students and anyone interested in mathematics. We will explore the steps required to find the value of ${ x }$ that satisfies the given equation. Additionally, we will discuss the implications of the negative sign under the square root and how it affects the solution. This comprehensive guide aims to provide a clear and detailed explanation, ensuring that you grasp the concepts and methods involved in solving this type of algebraic problem.

Understanding the Equation

To effectively solve the equation, we must first understand the components and their implications. The equation ${ \sqrt{-4x} = 100 }$ involves a square root, a negative sign, and a variable. Let's break down each part:

• Square Root: The square root function, denoted by ${ \sqrt{} }$, returns a value that, when multiplied by itself, equals the number under the root. For example, ${ \sqrt{9} = 3 }$ because ${ 3 \times 3 = 9 }$.
• Negative Sign: The presence of a negative sign under the square root introduces complexity. In the realm of real numbers, the square root of a negative number is undefined because no real number multiplied by itself yields a negative result. However, this leads us into the realm of imaginary numbers, which we will discuss further.
• Variable: The variable ${ x }$ represents the unknown value we are trying to find. Our goal is to isolate ${ x }$ on one side of the equation to determine its value.

Understanding these components is crucial for tackling the equation and arriving at the correct solution. The interaction between the negative sign and the square root will play a significant role in the solving process.

Step-by-Step Solution

To solve the equation ${ \sqrt{-4x} = 100 }$, we will follow a step-by-step approach to ensure clarity and accuracy.

Step 1: Squaring Both Sides

The first step in solving this equation is to eliminate the square root. We can do this by squaring both sides of the equation. This operation maintains the equality while removing the square root.

${ (\sqrt{-4x})^2 = 100^2 }$

Squaring both sides gives us:

${ -4x = 10000 }$

Step 2: Isolating the Variable

Now that we have eliminated the square root, we need to isolate the variable ${ x }$. This involves dividing both sides of the equation by the coefficient of ${ x }$, which is ${ -4 }$.

${ \frac{-4x}{-4} = \frac{10000}{-4} }$

Performing the division, we get:

${ x = -2500 }$

Step 3: Verification

It is essential to verify our solution by substituting the value of ${ x }$ back into the original equation to ensure it holds true.

${ \sqrt{-4(-2500)} = 100 }$

${ \sqrt{10000} = 100 }$

${ 100 = 100 }$

Since the equation holds true, our solution is correct.

Analyzing the Options

Now that we have found the solution, let's analyze the given options to determine the correct answer.

• A) x = –2500: This matches our solution, which we verified to be correct.
• B) x = –50: This is not the correct solution.
• C) x = –2.5: This is also not the correct solution.
• D) no solution: While it is important to consider cases where equations might not have solutions, in this case, we found a valid solution.

Therefore, the correct answer is A) x = –2500.

Implications of the Negative Sign

The negative sign under the square root in the original equation ${ \sqrt{-4x} = 100 }$ is a crucial aspect to consider. When we have a negative value under a square root, we are dealing with imaginary numbers. Imaginary numbers are multiples of the imaginary unit ${ i }$, where ${ i = \sqrt{-1} }$. However, in this specific problem, the negative sign is part of the term ${ -4x }$, and the variable ${ x }$ itself can be negative, which allows the expression under the square root to be positive.

When we substitute ${ x = -2500 }$ into ${ -4x }$, we get:

${ -4(-2500) = 10000 }$

The square root of 10000 is 100, which is a real number. This demonstrates that the negative sign, in this context, does not necessarily imply that we are dealing with imaginary numbers directly. Instead, it indicates that the value of ${ x }$ must be negative to make the expression under the square root positive.

This subtle but significant point is essential for understanding the nuances of algebraic equations involving square roots and negative numbers. It highlights the importance of considering the entire expression and how the variable interacts with the constants and operators involved.

Common Mistakes and How to Avoid Them

When solving equations like ${ \sqrt{-4x} = 100 }$, several common mistakes can occur. Being aware of these pitfalls can help you avoid them and solve the equation correctly.

Mistake 1: Forgetting to Square Both Sides

A common mistake is attempting to isolate the variable without first eliminating the square root. Squaring both sides is a crucial first step. Failing to do so will prevent you from correctly solving for ${ x }$. Remember, the order of operations is essential, and addressing the square root is paramount.

Mistake 2: Incorrectly Handling the Negative Sign

As discussed earlier, the negative sign under the square root can be confusing. Some students might prematurely conclude that there is no solution because they assume the square root of a negative number is undefined. However, the negative sign is part of the term ${ -4x }$, and the value of ${ x }$ can be negative, making the expression under the square root positive. Always consider the context and the potential for the variable to be negative.

Mistake 3: Arithmetic Errors

Simple arithmetic errors, such as mistakes in multiplication or division, can lead to incorrect solutions. For example, when dividing 10000 by -4, it's easy to make a mistake. Always double-check your calculations to ensure accuracy. It may be useful to use a calculator or write out the steps to minimize errors.

Mistake 4: Not Verifying the Solution

Failing to verify the solution by plugging it back into the original equation is a significant oversight. Verification is crucial to ensure that the solution is correct and that no extraneous solutions are included. Always take the time to substitute your answer back into the original equation to confirm its validity.

Mistake 5: Misunderstanding the Properties of Square Roots

Misunderstanding the properties of square roots can also lead to errors. For example, some students might incorrectly try to distribute the square root over terms, which is not a valid operation. Make sure you have a solid understanding of how square roots operate and the rules that govern them.

Real-World Applications

While solving equations like ${ \sqrt{-4x} = 100 }$ might seem purely academic, the concepts and skills involved have practical applications in various real-world scenarios. Understanding algebraic equations, square roots, and how to manipulate them is fundamental in fields such as physics, engineering, computer science, and finance.

Physics

In physics, square roots often appear in equations related to motion, energy, and waves. For example, the velocity of an object in free fall can be calculated using equations involving square roots. Solving for variables in these equations requires the same algebraic skills used in our example problem.

Engineering

Engineers use algebraic equations extensively in designing structures, circuits, and systems. Calculating stress, strain, and other physical properties often involves equations with square roots. The ability to accurately solve these equations is crucial for ensuring the safety and efficiency of engineering designs.

Computer Science

In computer science, square roots and algebraic equations are used in algorithms for graphics, simulations, and data analysis. For example, calculating distances and transformations in 3D graphics involves square roots. Understanding these mathematical concepts is essential for developing efficient and accurate computational methods.

Finance

In finance, square roots appear in formulas for calculating risk, return, and investment performance. The standard deviation, a key measure of risk, involves square roots. Financial analysts use these calculations to make informed decisions about investments and risk management.

Everyday Life

Even in everyday life, the skills learned from solving algebraic equations can be valuable. For example, calculating the dimensions of a garden, estimating travel time based on speed and distance, or determining the amount of material needed for a home improvement project can all involve algebraic thinking.

Conclusion

In conclusion, solving the equation ${ \sqrt{-4x} = 100 }$ involves several steps, including squaring both sides, isolating the variable, and verifying the solution. The correct solution is ${ x = -2500 }$. It is crucial to understand the implications of the negative sign under the square root and to avoid common mistakes by carefully following each step and verifying the answer. The skills and concepts learned in solving this type of equation have broad applications in various fields, making it an essential topic in mathematics education. By mastering these techniques, students can build a strong foundation for more advanced mathematical concepts and real-world problem-solving.