To solve the equation [tex]\((x-1)^2 = 50\)[/tex] and determine which of the given values of [tex]\(x\)[/tex] satisfy it, follow these steps:
1. Understand the given equation:
[tex]\[(x-1)^2 = 50\][/tex]
2. Isolate the squared term:
Take the square root on both sides of the equation:
[tex]\[
x - 1 = \pm \sqrt{50}
\][/tex]
3. Simplify the square root:
Recognize that [tex]\(\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}\)[/tex]:
[tex]\[
x - 1 = \pm 5\sqrt{2}
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
This yields two equations:
[tex]\[
x - 1 = 5\sqrt{2} \quad \text{and} \quad x - 1 = -5\sqrt{2}
\][/tex]
Solving each equation for [tex]\(x\)[/tex], we get:
[tex]\[
x = 1 + 5\sqrt{2} \quad \text{and} \quad x = 1 - 5\sqrt{2}
\][/tex]
5. Identify the valid solutions:
The solutions to the equation [tex]\((x-1)^2 = 50\)[/tex] are:
[tex]\[
x = 1 + 5\sqrt{2} \quad \text{and} \quad x = 1 - 5\sqrt{2}
\][/tex]
6. Compare with the given options:
The given options are:
- [tex]\(x = -49\)[/tex]
- [tex]\(x = 51\)[/tex]
- [tex]\(x = 1 + 5\sqrt{2}\)[/tex]
- [tex]\(x = 1 - 5\sqrt{2}\)[/tex]
7. Check which options match the solutions:
By comparing the solutions [tex]\(x = 1 + 5\sqrt{2}\)[/tex] and [tex]\(x = 1 - 5\sqrt{2}\)[/tex] with the given options, you can see:
- [tex]\(x = 1 + 5\sqrt{2}\)[/tex] matches one of our solutions.
- [tex]\(x = 1 - 5\sqrt{2}\)[/tex] matches the other solution.
- [tex]\(x = -49\)[/tex] does not match any of our solutions.
- [tex]\(x = 51\)[/tex] does not match any of our solutions.
Therefore, the correct values of [tex]\(x\)[/tex] that satisfy the equation [tex]\((x-1)^2 = 50\)[/tex] are:
[tex]\[
x = 1 + 5\sqrt{2} \quad \text{and} \quad x = 1 - 5\sqrt{2}
\][/tex]
