Answer :
Let's analyze each of the given equations to determine which one can be rewritten as [tex]\(x + 4 = x^2\)[/tex]:
1. [tex]\(\sqrt{x} + 2 = x\)[/tex]
- First, isolate the square root term: [tex]\(\sqrt{x} = x - 2\)[/tex].
- Square both sides to eliminate the square root: [tex]\(x = (x - 2)^2\)[/tex].
- Expand the right-hand side: [tex]\(x = x^2 - 4x + 4\)[/tex].
- Rearrange the equation: [tex]\(x^2 - 5x + 4 = 0\)[/tex].
- This does not match [tex]\(x^2 - x - 4 = 0\)[/tex], so the first equation does not satisfy the condition [tex]\(x + 4 = x^2\)[/tex].
2. [tex]\(\sqrt{x + 2} = x\)[/tex]
- Square both sides to eliminate the square root: [tex]\(x + 2 = x^2\)[/tex].
- Rearrange the equation: [tex]\(x^2 - x - 2 = 0\)[/tex].
- This does not match [tex]\(x^2 - x - 4 = 0\)[/tex], so the second equation does not satisfy the condition [tex]\(x + 4 = x^2\)[/tex].
3. [tex]\(\sqrt{x + 4} = x\)[/tex]
- Square both sides to eliminate the square root: [tex]\(x + 4 = x^2\)[/tex].
- Rearrange the equation: [tex]\(x^2 - x - 4 = 0\)[/tex].
- This matches the condition [tex]\(x + 4 = x^2\)[/tex], so the third equation satisfies the given condition.
4. [tex]\(\sqrt{x^2 + 16} = x\)[/tex]
- Square both sides to eliminate the square root: [tex]\(x^2 + 16 = x^2\)[/tex].
- Rearrange the equation: [tex]\(16 = 0\)[/tex].
- This is a contradiction and does not match [tex]\(x + 4 = x^2\)[/tex], so the fourth equation does not satisfy the condition.
Therefore, the equation that can be rewritten as [tex]\(x + 4 = x^2\)[/tex] is the third one:
[tex]\[ \boxed{3} \][/tex]
1. [tex]\(\sqrt{x} + 2 = x\)[/tex]
- First, isolate the square root term: [tex]\(\sqrt{x} = x - 2\)[/tex].
- Square both sides to eliminate the square root: [tex]\(x = (x - 2)^2\)[/tex].
- Expand the right-hand side: [tex]\(x = x^2 - 4x + 4\)[/tex].
- Rearrange the equation: [tex]\(x^2 - 5x + 4 = 0\)[/tex].
- This does not match [tex]\(x^2 - x - 4 = 0\)[/tex], so the first equation does not satisfy the condition [tex]\(x + 4 = x^2\)[/tex].
2. [tex]\(\sqrt{x + 2} = x\)[/tex]
- Square both sides to eliminate the square root: [tex]\(x + 2 = x^2\)[/tex].
- Rearrange the equation: [tex]\(x^2 - x - 2 = 0\)[/tex].
- This does not match [tex]\(x^2 - x - 4 = 0\)[/tex], so the second equation does not satisfy the condition [tex]\(x + 4 = x^2\)[/tex].
3. [tex]\(\sqrt{x + 4} = x\)[/tex]
- Square both sides to eliminate the square root: [tex]\(x + 4 = x^2\)[/tex].
- Rearrange the equation: [tex]\(x^2 - x - 4 = 0\)[/tex].
- This matches the condition [tex]\(x + 4 = x^2\)[/tex], so the third equation satisfies the given condition.
4. [tex]\(\sqrt{x^2 + 16} = x\)[/tex]
- Square both sides to eliminate the square root: [tex]\(x^2 + 16 = x^2\)[/tex].
- Rearrange the equation: [tex]\(16 = 0\)[/tex].
- This is a contradiction and does not match [tex]\(x + 4 = x^2\)[/tex], so the fourth equation does not satisfy the condition.
Therefore, the equation that can be rewritten as [tex]\(x + 4 = x^2\)[/tex] is the third one:
[tex]\[ \boxed{3} \][/tex]
