Given [tex]\((x-1)^2 = 50\)[/tex], select the values of [tex]\(x\)[/tex].

A. [tex]\(x = -49\)[/tex]
B. [tex]\(x = 51\)[/tex]
C. [tex]\(x = 1 + 5\sqrt{2}\)[/tex]
D. [tex]\(x = 1 - 5\sqrt{2}\)[/tex]



Answer :

To solve the equation [tex]\((x-1)^2 = 50\)[/tex], we can follow these steps:

1. Take the square root of both sides:

[tex]\[ (x-1)^2 = 50 \][/tex]
[tex]\[ x-1 = \pm \sqrt{50} \][/tex]

2. Simplify [tex]\(\sqrt{50}\)[/tex]:

[tex]\[ \sqrt{50} = \sqrt{25 \cdot 2} = 5 \sqrt{2} \][/tex]

3. So, we have two equations:

[tex]\[ x - 1 = 5 \sqrt{2} \quad \text{or} \quad x - 1 = -5 \sqrt{2} \][/tex]

4. Solve for [tex]\(x\)[/tex] in both cases:

[tex]\[ x - 1 = 5 \sqrt{2} \quad \Rightarrow \quad x = 1 + 5 \sqrt{2} \][/tex]
[tex]\[ x - 1 = -5 \sqrt{2} \quad \Rightarrow \quad x = 1 - 5 \sqrt{2} \][/tex]

Thus, 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]

From the given choices:
- [tex]\( x = -49 \)[/tex]
- [tex]\( x = 51 \)[/tex]

- [tex]\( x = 1 + 5 \sqrt{2} \)[/tex]
- [tex]\( x = 1 - 5 \sqrt{2} \)[/tex]

The correct values of [tex]\( x \)[/tex] are [tex]\( x = 1 + 5 \sqrt{2} \)[/tex] and [tex]\( x = 1 - 5 \sqrt{2} \)[/tex].