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]
