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Sagot :
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]
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]
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