Answered

Welcome to Westonci.ca, where you can find answers to all your questions from a community of experienced professionals. Connect with a community of experts ready to help you find solutions to your questions quickly and accurately. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.

What is the solution of [tex]\(x = 2 + \sqrt{x-2}\)[/tex]?

A. [tex]\(x = 2\)[/tex]

B. [tex]\(x = 3\)[/tex]

C. [tex]\(x = 2\)[/tex] or [tex]\(x = 3\)[/tex]

D. No solution

Sagot :

GinnyAnswer
To find the solution to the equation [tex]\( x = 2 + \sqrt{x - 2} \)[/tex], let's follow a detailed, step-by-step process.

1. Start with the given equation:
[tex]\[ x = 2 + \sqrt{x - 2} \][/tex]

2. Isolate the square root term:
[tex]\[ \sqrt{x - 2} = x - 2 \][/tex]

3. Square both sides to eliminate the square root:
[tex]\[ (\sqrt{x - 2})^2 = (x - 2)^2 \][/tex]
[tex]\[ x - 2 = (x - 2)^2 \][/tex]

4. Rewrite the equation:
[tex]\[ x - 2 = x^2 - 4x + 4 \][/tex]

5. Move all terms to one side to form a quadratic equation:
[tex]\[ 0 = x^2 - 4x + 4 - x + 2 \][/tex]
[tex]\[ 0 = x^2 - 5x + 6 \][/tex]

6. Factor the quadratic equation:
[tex]\[ 0 = (x - 2)(x - 3) \][/tex]

7. Set each factor to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \][/tex]
[tex]\[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \][/tex]

Hence, the solutions to the equation [tex]\( x = 2 + \sqrt{x - 2} \)[/tex] are [tex]\( x = 2 \)[/tex] and [tex]\( x = 3 \)[/tex].

The correct answer is:
[tex]\[ x = 2 \text{ or } x = 3 \][/tex]
We hope this information was helpful. Feel free to return anytime for more answers to your questions and concerns. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Thank you for visiting Westonci.ca, your go-to source for reliable answers. Come back soon for more expert insights.