To solve the equation [tex]\(\sqrt{x+14} + 2 = x\)[/tex], let's follow a detailed, step-by-step approach:
1. Isolate the square root term:
[tex]\[
\sqrt{x+14} = x - 2
\][/tex]
2. Square both sides to eliminate the square root:
[tex]\[
(\sqrt{x+14})^2 = (x - 2)^2
\][/tex]
This simplifies to:
[tex]\[
x + 14 = (x - 2)(x - 2)
\][/tex]
3. Expand the right-hand side:
[tex]\[
x + 14 = x^2 - 4x + 4
\][/tex]
4. Rearrange to form a quadratic equation:
[tex]\[
0 = x^2 - 5x - 10
\][/tex]
5. Solve the quadratic equation for [tex]\(x\)[/tex]:
This can be done using the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex], where [tex]\(a = 1\)[/tex], [tex]\(b = -5\)[/tex], and [tex]\(c = -10\)[/tex]. Plugging in these values:
[tex]\[
x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(-10)}}{2(1)}
\][/tex]
Simplifying further:
[tex]\[
x = \frac{5 \pm \sqrt{25 + 40}}{2}
\][/tex]
[tex]\[
x = \frac{5 \pm \sqrt{65}}{2}
\][/tex]
Therefore, the solutions to the quadratic equation are:
[tex]\[
x = \frac{5 + \sqrt{65}}{2} \quad \text{and} \quad x = \frac{5 - \sqrt{65}}{2}
\][/tex]
6. Verify the solutions to ensure they satisfy the original equation:
- For [tex]\(x = \frac{5 + \sqrt{65}}{2}\)[/tex]:
[tex]\[
\sqrt{\frac{5 + \sqrt{65}}{2} + 14} + 2 = \frac{5 + \sqrt{65}}{2}
\][/tex]
This simplifies correctly for the original equation, confirming it as a valid solution.
- For [tex]\(x = \frac{5 - \sqrt{65}}{2}\)[/tex]:
[tex]\[
\sqrt{\frac{5 - \sqrt{65}}{2} + 14} + 2 = \frac{5 - \sqrt{65}}{2}
\][/tex]
This, however, does not satisfy the original equation as it results in a non-real number under the square root.
Thus, the only valid solution to the equation [tex]\(\sqrt{x+14} + 2 = x\)[/tex] is:
[tex]\[
x = \frac{5 + \sqrt{65}}{2}
\][/tex]
Given the solutions provided in the question:
- [tex]\(x = -6\)[/tex]
- [tex]\(x = 1\)[/tex]
- Both [tex]\(x = -6\)[/tex] and [tex]\(x = 1\)[/tex]
None of these values match [tex]\(\frac{5 + \sqrt{65}}{2}\)[/tex], so the correct answer is:
[tex]\[
\boxed{\text{None of the above}}
\][/tex]
