To solve the equation [tex]\(\sqrt{x} + 6 = x\)[/tex], we can follow these steps:
1. Isolate the square root term:
[tex]\[\sqrt{x} + 6 = x\][/tex]
Subtract 6 from both sides to isolate the square root term:
[tex]\[\sqrt{x} = x - 6\][/tex]
2. Square both sides of the equation:
Squaring both sides will eliminate the square root:
[tex]\[(\sqrt{x})^2 = (x - 6)^2\][/tex]
This gives us:
[tex]\[x = (x - 6)^2\][/tex]
3. Expand the squared term:
Expand [tex]\((x - 6)^2\)[/tex]:
[tex]\[x = x^2 - 12x + 36\][/tex]
4. Rearrange the equation into standard quadratic form:
Move all terms to one side to set the equation to zero:
[tex]\[0 = x^2 - 12x + 36 - x\][/tex]
Simplify:
[tex]\[0 = x^2 - 13x + 36\][/tex]
5. Solve the quadratic equation:
Factor the quadratic expression:
[tex]\[x^2 - 13x + 36 = 0\][/tex]
To factor this, we look for two numbers that multiply to 36 and add to -13. These numbers are -9 and -4. So we can factor the quadratic as:
[tex]\[(x - 9)(x - 4) = 0\][/tex]
Therefore, the solutions are:
[tex]\[x = 9 \quad \text{or} \quad x = 4\][/tex]
6. Verify the solutions:
We need to check that these solutions satisfy the original equation:
- For [tex]\(x = 9\)[/tex]:
[tex]\[\sqrt{9} + 6 = 3 + 6 = 9\][/tex]
This is true.
- For [tex]\(x = 4\)[/tex]:
[tex]\[\sqrt{4} + 6 = 2 + 6 = 8\][/tex]
This is not true.
Thus, the valid solution that satisfies [tex]\(\sqrt{x} + 6 = x\)[/tex] is [tex]\(x = 9\)[/tex].
Therefore, the correct answer is:
B. 9
