Solve The Equation:$\sqrt{x-2} + 2 = X$

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Introduction

In this article, we will delve into the world of algebra and solve a seemingly complex equation involving a square root. The equation $\sqrt{x-2} + 2 = x$ may appear daunting at first, but with a step-by-step approach, we can break it down and find the solution. This equation is a great example of how algebraic manipulations can be used to isolate the variable and solve for its value.

Understanding the Equation

The given equation is $\sqrt{x-2} + 2 = x$. To solve this equation, we need to isolate the variable $x$. The equation involves a square root, which can be challenging to work with. However, we can use algebraic manipulations to simplify the equation and solve for $x$.

Step 1: Isolate the Square Root

The first step in solving the equation is to isolate the square root term. We can do this by subtracting 2 from both sides of the equation:

$$\sqrt{x-2} = x - 2$$

This step helps us to isolate the square root term, making it easier to work with.

Step 2: Square Both Sides

To eliminate the square root, we can square both sides of the equation:

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

This step is crucial in eliminating the square root, allowing us to work with a simpler equation.

Step 3: Expand and Simplify

Expanding and simplifying the equation, we get:

$$x - 2 = x^2 - 4x + 4$$

This step involves expanding the squared term and simplifying the equation.

Step 4: Rearrange the Equation

To isolate the variable $x$, we can rearrange the equation:

$$x^2 - 5x + 6 = 0$$

This step involves moving all the terms to one side of the equation, making it easier to solve.

Step 5: Factor the Quadratic Equation

The quadratic equation $x^2 - 5x + 6 = 0$ can be factored as:

$$(x - 2)(x - 3) = 0$$

This step involves factoring the quadratic equation, which helps us to find the solutions.

Step 6: Solve for $x$

To find the solutions, we can set each factor equal to zero:

$$x - 2 = 0 \quad \text{or} \quad x - 3 = 0$$

Solving for $x$, we get:

$$x = 2 \quad \text{or} \quad x = 3$$

This step involves solving for $x$ by setting each factor equal to zero.

Conclusion

In this article, we solved the equation $\sqrt{x-2} + 2 = x$ using a step-by-step approach. We isolated the square root term, squared both sides, expanded and simplified the equation, rearranged it, factored the quadratic equation, and finally solved for $x$. The solutions to the equation are $x = 2$ and $x = 3$. This equation is a great example of how algebraic manipulations can be used to solve complex equations involving square roots.

Final Answer

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

Additional Tips and Variations

• To verify the solutions, we can plug them back into the original equation.
• We can also use the quadratic formula to solve the quadratic equation $x^2 - 5x + 6 = 0$.
• The equation $\sqrt{x-2} + 2 = x$ can be modified to involve different types of equations, such as linear or polynomial equations.

Real-World Applications

The equation $\sqrt{x-2} + 2 = x$ has real-world applications in various fields, such as:

• Physics: The equation can be used to model the motion of objects under the influence of gravity.
• Engineering: The equation can be used to design and optimize systems, such as bridges or buildings.
• Economics: The equation can be used to model economic systems and make predictions about future trends.

Conclusion

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Introduction

In our previous article, we solved the equation $\sqrt{x-2} + 2 = x$ using a step-by-step approach. In this article, we will answer some frequently asked questions (FAQs) related to the equation and its solution.

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

A: The equation $\sqrt{x-2} + 2 = x$ is a quadratic equation that involves a square root. It is a type of equation that can be solved using algebraic manipulations and factoring.

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

A: To solve the equation, you can follow these steps:

1. Isolate the square root term by subtracting 2 from both sides of the equation.
2. Square both sides of the equation to eliminate the square root.
3. Expand and simplify the equation.
4. Rearrange the equation to isolate the variable $x$.
5. Factor the quadratic equation.
6. Solve for $x$ by setting each factor equal to zero.

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

A: The solutions to the equation are $x = 2$ and $x = 3$.

Q: How do I verify the solutions?

A: To verify the solutions, you can plug them back into the original equation. If the equation holds true for both solutions, then they are valid.

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

A: Yes, you can use the quadratic formula to solve the equation. The quadratic formula is:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

In this case, $a = 1$, $b = -5$, and $c = 6$. Plugging these values into the quadratic formula, you get:

$$x = \frac{5 \pm \sqrt{25 - 24}}{2}$$

Simplifying the expression, you get:

$$x = \frac{5 \pm 1}{2}$$

This gives you two possible solutions: $x = 3$ and $x = 2$.

Q: What are some real-world applications of the equation $\sqrt{x-2} + 2 = x$?

A: The equation $\sqrt{x-2} + 2 = x$ has real-world applications in various fields, such as:

• Physics: The equation can be used to model the motion of objects under the influence of gravity.
• Engineering: The equation can be used to design and optimize systems, such as bridges or buildings.
• Economics: The equation can be used to model economic systems and make predictions about future trends.

Q: Can I modify the equation $\sqrt{x-2} + 2 = x$ to involve different types of equations?

A: Yes, you can modify the equation to involve different types of equations, such as linear or polynomial equations. For example, you can replace the square root term with a linear term or a polynomial term.

Conclusion

In conclusion, the equation $\sqrt{x-2} + 2 = x$ is a quadratic equation that involves a square root. It can be solved using algebraic manipulations and factoring. The solutions to the equation are $x = 2$ and $x = 3$. This equation has real-world applications in various fields, and it can be modified to involve different types of equations.

Additional Resources

• For more information on solving quadratic equations, see our article on "Solving Quadratic Equations".
• For more information on real-world applications of quadratic equations, see our article on "Real-World Applications of Quadratic Equations".
• For more information on modifying quadratic equations, see our article on "Modifying Quadratic Equations".