Get the information you need quickly and easily with IDNLearn.com. Discover prompt and accurate answers from our community of experienced professionals.
Sagot :
To determine the solutions to the equation [tex]\((4x - 1)^2 = 11\)[/tex], we first need to solve this mathematical problem step-by-step.
1. Simplify the Equation:
Start by taking the square root of both sides of the equation:
[tex]\[ \sqrt{(4x - 1)^2} = \sqrt{11} \][/tex]
Which simplifies to:
[tex]\[ |4x - 1| = \sqrt{11} \][/tex]
2. Consider Both Cases for the Absolute Value:
The equation [tex]\(|4x - 1| = \sqrt{11}\)[/tex] leads to two possible equations:
[tex]\[ 4x - 1 = \sqrt{11} \][/tex]
[tex]\[ 4x - 1 = -\sqrt{11} \][/tex]
3. Solve Each Equation:
For the first equation:
[tex]\[ 4x - 1 = \sqrt{11} \][/tex]
Add 1 to both sides:
[tex]\[ 4x = \sqrt{11} + 1 \][/tex]
Divide by 4:
[tex]\[ x = \frac{\sqrt{11} + 1}{4} \][/tex]
For the second equation:
[tex]\[ 4x - 1 = -\sqrt{11} \][/tex]
Add 1 to both sides:
[tex]\[ 4x = -\sqrt{11} + 1 \][/tex]
Divide by 4:
[tex]\[ x = \frac{-\sqrt{11} + 1}{4} \][/tex]
4. Evaluate Possible Options:
Now, we are given the following options:
Option A: [tex]\( \frac{-\sqrt{11} + 1}{4} \)[/tex]
Option B: [tex]\( \sqrt{11} + \frac{1}{4} \)[/tex]
Option C: [tex]\( \frac{\sqrt{12}}{4} \)[/tex]
Option D: [tex]\( \frac{\sqrt{11} + 1}{4} \)[/tex]
Option E: [tex]\( -\frac{\sqrt{12}}{4} \)[/tex]
Option F: [tex]\( -\sqrt{11} + \frac{1}{4} \)[/tex]
We need to check which of these options match the calculated solutions [tex]\( \frac{\sqrt{11} + 1}{4} \)[/tex] and [tex]\( \frac{-\sqrt{11} + 1}{4} \)[/tex]. Let's evaluate them numerically for clarity.
- Option A: [tex]\( x = \frac{-\sqrt{11} + 1}{4} \approx -0.57915619758885 \)[/tex]
This is a match.
- Option B: [tex]\( x = \sqrt{11} + \frac{1}{4} \approx 3.33333333333333 \)[/tex]
This is not a match.
- Option C: [tex]\( x = \frac{\sqrt{12}}{4} \approx 0.86602540378443 \)[/tex]
This is not a match.
- Option D: [tex]\( x = \frac{\sqrt{11} + 1}{4} \approx 1.07915619758885 \)[/tex]
This is a match.
- Option E: [tex]\( x = -\frac{\sqrt{12}}{4} \approx -0.86602540378443 \)[/tex]
This is not a match.
- Option F: [tex]\( x = -\sqrt{11} + \frac{1}{4} \approx -0.57915619758885 \)[/tex]
This is a match.
Thus, the correct solutions are:
- Option A: [tex]\( x = \frac{-\sqrt{11} + 1}{4} \)[/tex]
- Option D: [tex]\( x = \frac{\sqrt{11} + 1}{4} \)[/tex]
- Option F: [tex]\( x = -\sqrt{11} + \frac{1}{4} \)[/tex]
1. Simplify the Equation:
Start by taking the square root of both sides of the equation:
[tex]\[ \sqrt{(4x - 1)^2} = \sqrt{11} \][/tex]
Which simplifies to:
[tex]\[ |4x - 1| = \sqrt{11} \][/tex]
2. Consider Both Cases for the Absolute Value:
The equation [tex]\(|4x - 1| = \sqrt{11}\)[/tex] leads to two possible equations:
[tex]\[ 4x - 1 = \sqrt{11} \][/tex]
[tex]\[ 4x - 1 = -\sqrt{11} \][/tex]
3. Solve Each Equation:
For the first equation:
[tex]\[ 4x - 1 = \sqrt{11} \][/tex]
Add 1 to both sides:
[tex]\[ 4x = \sqrt{11} + 1 \][/tex]
Divide by 4:
[tex]\[ x = \frac{\sqrt{11} + 1}{4} \][/tex]
For the second equation:
[tex]\[ 4x - 1 = -\sqrt{11} \][/tex]
Add 1 to both sides:
[tex]\[ 4x = -\sqrt{11} + 1 \][/tex]
Divide by 4:
[tex]\[ x = \frac{-\sqrt{11} + 1}{4} \][/tex]
4. Evaluate Possible Options:
Now, we are given the following options:
Option A: [tex]\( \frac{-\sqrt{11} + 1}{4} \)[/tex]
Option B: [tex]\( \sqrt{11} + \frac{1}{4} \)[/tex]
Option C: [tex]\( \frac{\sqrt{12}}{4} \)[/tex]
Option D: [tex]\( \frac{\sqrt{11} + 1}{4} \)[/tex]
Option E: [tex]\( -\frac{\sqrt{12}}{4} \)[/tex]
Option F: [tex]\( -\sqrt{11} + \frac{1}{4} \)[/tex]
We need to check which of these options match the calculated solutions [tex]\( \frac{\sqrt{11} + 1}{4} \)[/tex] and [tex]\( \frac{-\sqrt{11} + 1}{4} \)[/tex]. Let's evaluate them numerically for clarity.
- Option A: [tex]\( x = \frac{-\sqrt{11} + 1}{4} \approx -0.57915619758885 \)[/tex]
This is a match.
- Option B: [tex]\( x = \sqrt{11} + \frac{1}{4} \approx 3.33333333333333 \)[/tex]
This is not a match.
- Option C: [tex]\( x = \frac{\sqrt{12}}{4} \approx 0.86602540378443 \)[/tex]
This is not a match.
- Option D: [tex]\( x = \frac{\sqrt{11} + 1}{4} \approx 1.07915619758885 \)[/tex]
This is a match.
- Option E: [tex]\( x = -\frac{\sqrt{12}}{4} \approx -0.86602540378443 \)[/tex]
This is not a match.
- Option F: [tex]\( x = -\sqrt{11} + \frac{1}{4} \approx -0.57915619758885 \)[/tex]
This is a match.
Thus, the correct solutions are:
- Option A: [tex]\( x = \frac{-\sqrt{11} + 1}{4} \)[/tex]
- Option D: [tex]\( x = \frac{\sqrt{11} + 1}{4} \)[/tex]
- Option F: [tex]\( x = -\sqrt{11} + \frac{1}{4} \)[/tex]
Your presence in our community is highly appreciated. Keep sharing your insights and solutions. Together, we can build a rich and valuable knowledge resource for everyone. IDNLearn.com is your reliable source for answers. We appreciate your visit and look forward to assisting you again soon.
