What Is The Solution Of X = 2 + X − 2 X = 2 + \sqrt{x - 2} X = 2 + X − 2 ​ ?A. X = 2 X = 2 X = 2 B. X = 3 X = 3 X = 3 C. No Solution

Introduction

Mathematical equations are an essential part of mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific equation, $x = 2 + \sqrt{x - 2}$, and explore the different methods and techniques used to find its solution.

Understanding the Equation

The given equation is a quadratic equation, which is a polynomial equation of degree two. It has the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. In this case, the equation is $x = 2 + \sqrt{x - 2}$, which can be rewritten as $x - 2 = \sqrt{x - 2}$.

Isolating the Square Root

To solve the equation, we need to isolate the square root term. We can do this by squaring both sides of the equation. Squaring both sides gives us $(x - 2)^2 = x - 2$.

Expanding the Left Side

Expanding the left side of the equation gives us $x^2 - 4x + 4 = x - 2$.

Simplifying the Equation

Simplifying the equation gives us $x^2 - 5x + 6 = 0$.

Factoring the Quadratic

The quadratic equation $x^2 - 5x + 6 = 0$ can be factored as $(x - 2)(x - 3) = 0$.

Finding the Solutions

To find the solutions, we need to set each factor equal to zero and solve for $x$. Setting $x - 2 = 0$ gives us $x = 2$, and setting $x - 3 = 0$ gives us $x = 3$.

Checking the Solutions

We need to check if both solutions satisfy the original equation. Substituting $x = 2$ into the original equation gives us $2 = 2 + \sqrt{2 - 2}$, which simplifies to $2 = 2$. This is true, so $x = 2$ is a solution. Substituting $x = 3$ into the original equation gives us $3 = 2 + \sqrt{3 - 2}$, which simplifies to $3 = 2 + 1$. This is also true, so $x = 3$ is a solution.

Conclusion

In conclusion, the solution to the equation $x = 2 + \sqrt{x - 2}$ is $x = 2$ and $x = 3$. Both solutions satisfy the original equation, and they can be found by factoring the quadratic equation and solving for $x$.

Discussion

The equation $x = 2 + \sqrt{x - 2}$ is a quadratic equation that can be solved using various methods, including factoring and the quadratic formula. The solutions to the equation are $x = 2$ and $x = 3$, which can be found by setting each factor equal to zero and solving for $x$. This equation is a great example of how mathematical equations can be used to model real-world problems and how solving them can provide valuable insights and solutions.

Final Answer

The final answer is: $\boxed{2, 3}$

Introduction

In our previous article, we explored the solution to the equation $x = 2 + \sqrt{x - 2}$. In this article, we will answer some frequently asked questions (FAQs) about the equation and provide additional insights and explanations.

Q: What is the significance of the equation $x = 2 + \sqrt{x - 2}$?

A: The equation $x = 2 + \sqrt{x - 2}$ is a quadratic equation that can be used to model real-world problems. It has applications in various fields, including mathematics, physics, and engineering.

Q: How do I solve the equation $x = 2 + \sqrt{x - 2}$?

A: To solve the equation, you can use various methods, including factoring and the quadratic formula. We have already shown how to solve the equation using factoring in our previous article.

Q: What are the solutions to the equation $x = 2 + \sqrt{x - 2}$?

A: The solutions to the equation are $x = 2$ and $x = 3$. Both solutions satisfy the original equation and can be found by setting each factor equal to zero and solving for $x$.

Q: Why do we need to check the solutions?

A: We need to check the solutions to ensure that they satisfy the original equation. This is an important step in solving equations, as it helps us to verify that our solutions are correct.

Q: Can I use the quadratic formula to solve the equation $x = 2 + \sqrt{x - 2}$?

A: Yes, you can use the quadratic formula to solve the equation. The quadratic formula is a general method for solving quadratic equations, and it can be used to find the solutions to the equation $x = 2 + \sqrt{x - 2}$.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that can be used to find the solutions to quadratic equations. It is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a$, $b$, and $c$ are the coefficients of the quadratic equation.

Q: How do I apply the quadratic formula to the equation $x = 2 + \sqrt{x - 2}$?

A: To apply the quadratic formula, you need to rewrite the equation in the standard form $ax^2 + bx + c = 0$. In this case, the equation can be rewritten as $x^2 - 5x + 6 = 0$. Then, you can plug in the values of $a$, $b$, and $c$ into the quadratic formula and simplify to find the solutions.

Q: What are some common mistakes to avoid when solving the equation $x = 2 + \sqrt{x - 2}$?

A: Some common mistakes to avoid when solving the equation include:

• Not checking the solutions to ensure that they satisfy the original equation
• Not using the correct method to solve the equation (e.g., factoring or the quadratic formula)
• Not simplifying the equation correctly
• Not considering the possibility of complex solutions

Conclusion

In conclusion, the equation $x = 2 + \sqrt{x - 2}$ is a quadratic equation that can be solved using various methods, including factoring and the quadratic formula. We have answered some frequently asked questions (FAQs) about the equation and provided additional insights and explanations. By following the steps outlined in this article, you can solve the equation and gain a deeper understanding of quadratic equations.

Final Answer

The final answer is: $\boxed{2, 3}$