IDNLearn.com: Your trusted platform for finding reliable answers. Discover detailed answers to your questions with our extensive database of expert knowledge.
Sagot :
To solve the given radical equation:
[tex]\[ x + 1 = \sqrt{-6x - 6} \][/tex]
we need to follow several steps methodically to find the solutions and check for any extraneous ones.
### Step-by-Step Solution
1. Isolate the Radical Expression:
[tex]\[ x + 1 = \sqrt{-6x - 6} \][/tex]
2. Square Both Sides to eliminate the square root. This gives:
[tex]\[ (x + 1)^2 = (\sqrt{-6x - 6})^2 \][/tex]
Simplifying, we get:
[tex]\[ (x + 1)^2 = -6x - 6 \][/tex]
3. Expand the Left Side:
[tex]\[ x^2 + 2x + 1 = -6x - 6 \][/tex]
4. Move All Terms to One Side to set the equation to zero:
[tex]\[ x^2 + 2x + 1 + 6x + 6 = 0 \][/tex]
Simplifying further:
[tex]\[ x^2 + 8x + 7 = 0 \][/tex]
5. Solve the Quadratic Equation using the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex] where [tex]\(a = 1\)[/tex], [tex]\(b = 8\)[/tex], and [tex]\(c = 7\)[/tex]:
[tex]\[ x = \frac{-8 \pm \sqrt{8^2 - 4 \cdot 1 \cdot 7}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{-8 \pm \sqrt{64 - 28}}{2} \][/tex]
[tex]\[ x = \frac{-8 \pm \sqrt{36}}{2} \][/tex]
[tex]\[ x = \frac{-8 \pm 6}{2} \][/tex]
This gives two solutions:
[tex]\[ x = \frac{-8 + 6}{2} = \frac{-2}{2} = -1 \][/tex]
[tex]\[ x = \frac{-8 - 6}{2} = \frac{-14}{2} = -7 \][/tex]
6. Check Each Solution for Extraneous Solutions by substituting them back into the original equation [tex]\(x + 1 = \sqrt{-6x - 6}\)[/tex]:
- For [tex]\(x = -1\)[/tex]:
[tex]\[ -1 + 1 = \sqrt{-6(-1) - 6} \][/tex]
[tex]\[ 0 = \sqrt{6 - 6} \][/tex]
[tex]\[ 0 = \sqrt{0} \][/tex]
[tex]\[ 0 = 0 \, \text{(True)} \][/tex]
Hence, [tex]\(x = -1\)[/tex] is a valid solution.
- For [tex]\(x = -7\)[/tex]:
[tex]\[ -7 + 1 = \sqrt{-6(-7) - 6} \][/tex]
[tex]\[ -6 = \sqrt{42 - 6} \][/tex]
[tex]\[ -6 = \sqrt{36} \][/tex]
[tex]\[ -6 \neq 6 \, \text{(False)} \][/tex]
Hence, [tex]\(x = -7\)[/tex] is an extraneous solution.
### Conclusion
The solution [tex]\(x = -1\)[/tex] is a true solution to the equation [tex]\(x + 1 = \sqrt{-6x - 6}\)[/tex]. The solution [tex]\(x = -7\)[/tex] is an extraneous solution. Therefore:
- The solution [tex]\(x = -1\)[/tex] is not an extraneous solution.
- The solution [tex]\(x = -7\)[/tex] is an extraneous solution.
So the correct statement is:
- The solution [tex]\(x = -7\)[/tex] is an extraneous solution.
[tex]\[ x + 1 = \sqrt{-6x - 6} \][/tex]
we need to follow several steps methodically to find the solutions and check for any extraneous ones.
### Step-by-Step Solution
1. Isolate the Radical Expression:
[tex]\[ x + 1 = \sqrt{-6x - 6} \][/tex]
2. Square Both Sides to eliminate the square root. This gives:
[tex]\[ (x + 1)^2 = (\sqrt{-6x - 6})^2 \][/tex]
Simplifying, we get:
[tex]\[ (x + 1)^2 = -6x - 6 \][/tex]
3. Expand the Left Side:
[tex]\[ x^2 + 2x + 1 = -6x - 6 \][/tex]
4. Move All Terms to One Side to set the equation to zero:
[tex]\[ x^2 + 2x + 1 + 6x + 6 = 0 \][/tex]
Simplifying further:
[tex]\[ x^2 + 8x + 7 = 0 \][/tex]
5. Solve the Quadratic Equation using the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex] where [tex]\(a = 1\)[/tex], [tex]\(b = 8\)[/tex], and [tex]\(c = 7\)[/tex]:
[tex]\[ x = \frac{-8 \pm \sqrt{8^2 - 4 \cdot 1 \cdot 7}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{-8 \pm \sqrt{64 - 28}}{2} \][/tex]
[tex]\[ x = \frac{-8 \pm \sqrt{36}}{2} \][/tex]
[tex]\[ x = \frac{-8 \pm 6}{2} \][/tex]
This gives two solutions:
[tex]\[ x = \frac{-8 + 6}{2} = \frac{-2}{2} = -1 \][/tex]
[tex]\[ x = \frac{-8 - 6}{2} = \frac{-14}{2} = -7 \][/tex]
6. Check Each Solution for Extraneous Solutions by substituting them back into the original equation [tex]\(x + 1 = \sqrt{-6x - 6}\)[/tex]:
- For [tex]\(x = -1\)[/tex]:
[tex]\[ -1 + 1 = \sqrt{-6(-1) - 6} \][/tex]
[tex]\[ 0 = \sqrt{6 - 6} \][/tex]
[tex]\[ 0 = \sqrt{0} \][/tex]
[tex]\[ 0 = 0 \, \text{(True)} \][/tex]
Hence, [tex]\(x = -1\)[/tex] is a valid solution.
- For [tex]\(x = -7\)[/tex]:
[tex]\[ -7 + 1 = \sqrt{-6(-7) - 6} \][/tex]
[tex]\[ -6 = \sqrt{42 - 6} \][/tex]
[tex]\[ -6 = \sqrt{36} \][/tex]
[tex]\[ -6 \neq 6 \, \text{(False)} \][/tex]
Hence, [tex]\(x = -7\)[/tex] is an extraneous solution.
### Conclusion
The solution [tex]\(x = -1\)[/tex] is a true solution to the equation [tex]\(x + 1 = \sqrt{-6x - 6}\)[/tex]. The solution [tex]\(x = -7\)[/tex] is an extraneous solution. Therefore:
- The solution [tex]\(x = -1\)[/tex] is not an extraneous solution.
- The solution [tex]\(x = -7\)[/tex] is an extraneous solution.
So the correct statement is:
- The solution [tex]\(x = -7\)[/tex] is an extraneous solution.
Thank you for joining our conversation. Don't hesitate to return anytime to find answers to your questions. Let's continue sharing knowledge and experiences! Discover insightful answers at IDNLearn.com. We appreciate your visit and look forward to assisting you again.