What Is The Solution Of $x = 2 + \sqrt{x - 2}$?A. $x = 2$ B. $ X = 3 X = 3 X = 3 [/tex] C. $x = 2$ Or $x = 3$ D. No Solution

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Introduction

In mathematics, solving equations is a crucial aspect of problem-solving. Equations are statements that express the equality of two mathematical expressions. Solving an equation involves finding the value or values of the variable that make the equation true. In this article, we will focus on solving a specific equation, $x = 2 + \sqrt{x - 2}$, and explore the possible solutions.

Understanding the Equation

The given equation is $x = 2 + \sqrt{x - 2}$. To solve this equation, we need to isolate the variable xx. The equation involves a square root term, which can be challenging to handle. However, we can start by isolating the square root term.

Isolating the Square Root Term

We can rewrite the equation as:

x−2=x−2x - 2 = \sqrt{x - 2}

This step involves subtracting 2 from both sides of the equation, which helps to isolate the square root term.

Squaring Both Sides

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

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

This step involves squaring both sides of the equation, which helps to eliminate the square root term.

Expanding the Squared Terms

We can expand the squared terms on both sides of the equation:

x2−4x+4=x−2x^2 - 4x + 4 = x - 2

This step involves expanding the squared terms on both sides of the equation.

Simplifying the Equation

We can simplify the equation by combining like terms:

x2−5x+6=0x^2 - 5x + 6 = 0

This step involves combining like terms on the left-hand side of the equation.

Factoring the Quadratic Equation

We can factor the quadratic equation:

(x−2)(x−3)=0(x - 2)(x - 3) = 0

This step involves factoring the quadratic equation.

Finding the Solutions

We can find the solutions by setting each factor equal to zero:

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

This step involves setting each factor equal to zero.

Solving for x

We can solve for xx by adding 2 to both sides of the first equation and adding 3 to both sides of the second equation:

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

This step involves solving for xx.

Conclusion

In conclusion, the solution to the given equation $x = 2 + \sqrt{x - 2}$ is x=2x = 2 or x=3x = 3. This solution is obtained by isolating the square root term, squaring both sides, expanding the squared terms, simplifying the equation, factoring the quadratic equation, and finding the solutions.

Discussion

The given equation is a quadratic equation, and the solution involves factoring the quadratic equation. The solution is x=2x = 2 or x=3x = 3, which means that there are two possible solutions to the equation.

Final Answer

Q&A: Solving the Equation

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

Q: How do we start solving the equation? A: We start by isolating the square root term. We can rewrite the equation as $x - 2 = \sqrt{x - 2}$.

Q: Why do we isolate the square root term? A: We isolate the square root term to eliminate it. By isolating the square root term, we can square both sides of the equation to eliminate the square root.

Q: What happens when we square both sides of the equation? A: When we square both sides of the equation, we get $(x - 2)^2 = (\sqrt{x - 2})^2$.

Q: How do we simplify the equation after squaring both sides? A: We simplify the equation by expanding the squared terms on both sides of the equation. This gives us $x^2 - 4x + 4 = x - 2$.

Q: How do we further simplify the equation? A: We further simplify the equation by combining like terms. This gives us $x^2 - 5x + 6 = 0$.

Q: What type of equation do we have now? A: We have a quadratic equation.

Q: How do we solve the quadratic equation? A: We solve the quadratic equation by factoring it. This gives us $(x - 2)(x - 3) = 0$.

Q: What are the solutions to the equation? A: The solutions to the equation are $x = 2$ or $x = 3$.

Q: Why do we have two solutions? A: We have two solutions because the quadratic equation has two factors, $(x - 2)$ and $(x - 3)$. Each factor can be equal to zero, which gives us two possible values for xx.

Q: What is the final answer? A: The final answer is $x = 2$ or $x = 3$.

Common Mistakes to Avoid

  • Not isolating the square root term before squaring both sides of the equation.
  • Not expanding the squared terms on both sides of the equation after squaring both sides.
  • Not combining like terms after expanding the squared terms.
  • Not factoring the quadratic equation after combining like terms.
  • Not setting each factor equal to zero after factoring the quadratic equation.

Tips and Tricks

  • Always isolate the square root term before squaring both sides of the equation.
  • Always expand the squared terms on both sides of the equation after squaring both sides.
  • Always combine like terms after expanding the squared terms.
  • Always factor the quadratic equation after combining like terms.
  • Always set each factor equal to zero after factoring the quadratic equation.

Conclusion

In conclusion, solving the equation $x = 2 + \sqrt{x - 2}$ involves isolating the square root term, squaring both sides, expanding the squared terms, combining like terms, factoring the quadratic equation, and setting each factor equal to zero. The final answer is $x = 2$ or $x = 3$.