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]
