To determine which equation can be rewritten as [tex]\(x + 4 = x^2\)[/tex], we'll solve each equation step-by-step and see which one matches [tex]\(x + 4 = x^2\)[/tex].
### Equation 1: [tex]\(\sqrt{x} + 2 = x\)[/tex]
1. Start with the given equation:
[tex]\[ \sqrt{x} + 2 = x \][/tex]
2. Isolate the square root term:
[tex]\[ \sqrt{x} = x - 2 \][/tex]
3. Square both sides to eliminate the square root:
[tex]\[ (\sqrt{x})^2 = (x - 2)^2 \][/tex]
[tex]\[ x = x^2 - 4x + 4 \][/tex]
4. Rearrange the equation to standard quadratic form:
[tex]\[ 0 = x^2 - 5x + 4 \][/tex]
[tex]\[ x^2 - 5x + 4 = 0 \][/tex]
This equation is not equivalent to [tex]\(x + 4 = x^2\)[/tex].
### Equation 2: [tex]\(\sqrt{x+2} = x\)[/tex]
1. Start with the given equation:
[tex]\[ \sqrt{x+2} = x \][/tex]
2. Square both sides to eliminate the square root:
[tex]\[ (\sqrt{x+2})^2 = x^2 \][/tex]
[tex]\[ x + 2 = x^2 \][/tex]
3. Rearrange the equation to standard quadratic form:
[tex]\[ x^2 - x - 2 = 0 \][/tex]
This equation is not equivalent to [tex]\(x + 4 = x^2\)[/tex].
### Equation 3: [tex]\(\sqrt{x+4} = x\)[/tex]
1. Start with the given equation:
[tex]\[ \sqrt{x+4} = x \][/tex]
2. Square both sides to eliminate the square root:
[tex]\[ (\sqrt{x+4})^2 = x^2 \][/tex]
[tex]\[ x + 4 = x^2 \][/tex]
This equation is exactly [tex]\(x + 4 = x^2\)[/tex].
### Equation 4: [tex]\(\sqrt{x^2 + 16} = x\)[/tex]
1. Start with the given equation:
[tex]\[ \sqrt{x^2 + 16} = x \][/tex]
2. Square both sides to eliminate the square root:
[tex]\[ (\sqrt{x^2 + 16})^2 = x^2 \][/tex]
[tex]\[ x^2 + 16 = x^2 \][/tex]
This results in:
[tex]\[ 16 = 0 \][/tex]
This is a contradiction and cannot be true for any [tex]\(x\)[/tex].
Therefore, the equation that can be rewritten as [tex]\(x + 4 = x^2\)[/tex] is:
[tex]\[ \sqrt{x + 4} = x \][/tex]
