Let's solve the equation [tex]\((x-1)^2 = 50\)[/tex] step-by-step to find the correct values for [tex]\( x \)[/tex].
First, we start by expanding the left side of the equation:
[tex]\[
(x - 1)^2 = 50
\][/tex]
To solve this, we need to take the square root of both sides:
[tex]\[
x - 1 = \pm \sqrt{50}
\][/tex]
Simplifying [tex]\(\sqrt{50}\)[/tex]:
[tex]\[
\sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}
\][/tex]
Thus, we have two possible equations:
[tex]\[
x - 1 = 5\sqrt{2} \quad \text{or} \quad x - 1 = -5\sqrt{2}
\][/tex]
Solving for [tex]\( x \)[/tex] in each case:
[tex]\[
x = 1 + 5\sqrt{2} \quad \text{or} \quad x = 1 - 5\sqrt{2}
\][/tex]
So, the solutions for [tex]\( x \)[/tex] are:
[tex]\[
x = 1 + 5\sqrt{2} \quad \text{and} \quad x = 1 - 5\sqrt{2}
\][/tex]
Next, let's compare these solutions to the given options:
1. [tex]\( x = -49 \)[/tex]
2. [tex]\( x = 51 \)[/tex]
3. [tex]\( x = 1 + 5\sqrt{2} \)[/tex]
4. [tex]\( x = 1 - 5\sqrt{2} \)[/tex]
We already identified the solutions to be [tex]\( x = 1 + 5\sqrt{2} \)[/tex] and [tex]\( x = 1 - 5\sqrt{2} \)[/tex].
Checking each option:
1. [tex]\( x = -49 \)[/tex]: This is not a solution.
2. [tex]\( x = 51 \)[/tex]: This is not a solution.
3. [tex]\( x = 1 + 5\sqrt{2} \)[/tex]: This is a solution.
4. [tex]\( x = 1 - 5\sqrt{2} \)[/tex]: This is a solution.
Thus, the correct values for [tex]\( x \)[/tex] are:
[tex]\[
x = 1 + 5\sqrt{2} \quad \text{and} \quad x = 1 - 5\sqrt{2}
\][/tex]
