Solve The Equation:${ x - 4 = (\sqrt{x} - 2)(\sqrt{x} + 2) }$

Introduction

In this article, we will delve into the world of mathematics and explore a fascinating equation that involves square roots. The equation in question is $x - 4 = (\sqrt{x} - 2)(\sqrt{x} + 2)$. Our goal is to solve this equation and uncover the values of $x$ that satisfy it. We will break down the solution into manageable steps, making it easy to follow and understand.

Step 1: Simplifying the Equation

To begin, let's simplify the equation by multiplying the terms on the right-hand side. We can use the difference of squares formula, which states that $(a-b)(a+b) = a^2 - b^2$. Applying this formula to our equation, we get:

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

Simplifying further, we have:

$x - 4 = x - 4$

At first glance, it may seem like the equation is true for all values of $x$. However, this is not the case. We need to be careful and consider the domain of the square root function.

Step 2: Considering the Domain of the Square Root Function

The square root function is defined only for non-negative real numbers. This means that $\sqrt{x}$ is only defined when $x \geq 0$. Therefore, we need to restrict our solution to values of $x$ that satisfy this condition.

Step 3: Solving for $x$

Now that we have simplified the equation and considered the domain of the square root function, we can solve for $x$. We can start by adding $4$ to both sides of the equation:

$x = x - 4 + 4$

Simplifying further, we have:

$x = x$

This equation is true for all values of $x$, but we need to remember that we are only interested in values of $x$ that satisfy the condition $x \geq 0$.

Step 4: Finding the Solution

To find the solution, we need to consider the values of $x$ that satisfy the condition $x \geq 0$. We can start by plugging in some values of $x$ to see if they satisfy the equation.

Let's try plugging in $x = 0$:

$0 - 4 = (\sqrt{0} - 2)(\sqrt{0} + 2)$

Simplifying, we have:

$-4 = (-2)(2)$

This is not true, so $x = 0$ is not a solution.

Let's try plugging in $x = 1$:

$1 - 4 = (\sqrt{1} - 2)(\sqrt{1} + 2)$

Simplifying, we have:

$-3 = (-1)(3)$

This is not true, so $x = 1$ is not a solution.

Let's try plugging in $x = 4$:

$4 - 4 = (\sqrt{4} - 2)(\sqrt{4} + 2)$

Simplifying, we have:

$0 = (2 - 2)(2 + 2)$

This is true, so $x = 4$ is a solution.

Conclusion

In this article, we have solved the equation $x - 4 = (\sqrt{x} - 2)(\sqrt{x} + 2)$. We have broken down the solution into manageable steps, simplifying the equation and considering the domain of the square root function. We have found that the solution is $x = 4$, and we have verified this solution by plugging it back into the original equation.

Final Answer

The final answer is $\boxed{4}$.

Additional Resources

If you are interested in learning more about solving equations and inequalities, I recommend checking out the following resources:

• Khan Academy: Solving Equations and Inequalities
• Mathway: Solving Equations and Inequalities
• Wolfram Alpha: Solving Equations and Inequalities

These resources provide a wealth of information and practice problems to help you improve your skills in solving equations and inequalities.

Common Mistakes

When solving equations and inequalities, it's easy to make mistakes. Here are some common mistakes to watch out for:

• Not considering the domain of the square root function
• Not simplifying the equation properly
• Not verifying the solution by plugging it back into the original equation

By being aware of these common mistakes, you can avoid them and ensure that your solutions are accurate and reliable.

Real-World Applications

Solving equations and inequalities has many real-world applications. Here are a few examples:

• Physics: Solving equations and inequalities is essential in physics, where you need to solve equations to describe the motion of objects and predict their behavior.
• Engineering: Solving equations and inequalities is also essential in engineering, where you need to solve equations to design and optimize systems.
• Economics: Solving equations and inequalities is also essential in economics, where you need to solve equations to model economic systems and make predictions about future trends.

Introduction

In our previous article, we solved the equation $x - 4 = (\sqrt{x} - 2)(\sqrt{x} + 2)$. We broke down the solution into manageable steps, simplifying the equation and considering the domain of the square root function. We found that the solution is $x = 4$, and we verified this solution by plugging it back into the original equation.

In this article, we will answer some frequently asked questions about solving the equation. We will also provide additional resources and tips to help you improve your skills in solving equations and inequalities.

Q&A

Q: What is the domain of the square root function?

A: The domain of the square root function is all non-negative real numbers, i.e., $x \geq 0$.

Q: How do I simplify the equation?

A: To simplify the equation, you can use the difference of squares formula, which states that $(a-b)(a+b) = a^2 - b^2$. You can also use other algebraic manipulations, such as combining like terms and factoring.

Q: Why do I need to consider the domain of the square root function?

A: You need to consider the domain of the square root function because it is only defined for non-negative real numbers. If you plug in a negative value for $x$, the square root function will be undefined.

Q: How do I verify the solution?

A: To verify the solution, you can plug it back into the original equation and check if it is true. You can also use other methods, such as graphing or numerical methods, to verify the solution.

Q: What are some common mistakes to watch out for?

A: Some common mistakes to watch out for when solving equations and inequalities include:

• Not considering the domain of the square root function
• Not simplifying the equation properly
• Not verifying the solution by plugging it back into the original equation

Q: How do I apply solving equations and inequalities to real-world problems?

A: Solving equations and inequalities has many real-world applications, including physics, engineering, and economics. You can apply the skills you learn in solving equations and inequalities to a wide range of problems, such as:

• Modeling the motion of objects in physics
• Designing and optimizing systems in engineering
• Modeling economic systems and making predictions about future trends

Q: What are some additional resources for learning about solving equations and inequalities?

A: Some additional resources for learning about solving equations and inequalities include:

• Khan Academy: Solving Equations and Inequalities
• Mathway: Solving Equations and Inequalities
• Wolfram Alpha: Solving Equations and Inequalities

These resources provide a wealth of information and practice problems to help you improve your skills in solving equations and inequalities.

Tips and Tricks

Here are some tips and tricks to help you improve your skills in solving equations and inequalities:

• Practice, practice, practice! The more you practice, the more comfortable you will become with solving equations and inequalities.
• Use algebraic manipulations, such as combining like terms and factoring, to simplify the equation.
• Consider the domain of the square root function and other functions that may be involved in the equation.
• Verify the solution by plugging it back into the original equation.
• Use graphing or numerical methods to verify the solution.

Conclusion

Solving equations and inequalities is an essential skill in mathematics and has many real-world applications. By mastering the skills of solving equations and inequalities, you can apply them to a wide range of problems and make a positive impact in your community. We hope this article has been helpful in answering your questions and providing additional resources and tips to help you improve your skills in solving equations and inequalities.

Final Answer

The final answer is $\boxed{4}$.

Additional Resources

If you are interested in learning more about solving equations and inequalities, we recommend checking out the following resources:

• Khan Academy: Solving Equations and Inequalities
• Mathway: Solving Equations and Inequalities
• Wolfram Alpha: Solving Equations and Inequalities

These resources provide a wealth of information and practice problems to help you improve your skills in solving equations and inequalities.