To find the value of [tex]\( x \)[/tex] from the given equation [tex]\(\sqrt{x} - 1 = \sqrt{x - 3}\)[/tex], follow these steps:
1. Isolate one of the square roots:
Given equation:
[tex]\[
\sqrt{x} - 1 = \sqrt{x - 3}
\][/tex]
Add 1 to both sides to isolate [tex]\(\sqrt{x}\)[/tex]:
[tex]\[
\sqrt{x} = \sqrt{x - 3} + 1
\][/tex]
2. Square both sides to eliminate the square roots:
By squaring both sides, we can deal with the radicals:
[tex]\[
(\sqrt{x})^2 = (\sqrt{x - 3} + 1)^2
\][/tex]
This simplifies to:
[tex]\[
x = (\sqrt{x - 3})^2 + 2\sqrt{x - 3} \cdot 1 + 1
\][/tex]
[tex]\[
x = (x - 3) + 2\sqrt{x - 3} + 1
\][/tex]
[tex]\[
x = x - 3 + 2\sqrt{x - 3} + 1
\][/tex]
3. Simplify the equation:
Combine like terms:
[tex]\[
x = x - 2 + 2\sqrt{x - 3}
\][/tex]
Subtract [tex]\( x \)[/tex] from both sides to isolate the radical expression:
[tex]\[
0 = -2 + 2\sqrt{x - 3}
\][/tex]
4. Solve for the square root term:
Add 2 to both sides:
[tex]\[
2 = 2\sqrt{x - 3}
\][/tex]
Divide by 2:
[tex]\[
1 = \sqrt{x - 3}
\][/tex]
5. Square both sides again to eliminate the remaining square root:
[tex]\[
1^2 = (\sqrt{x - 3})^2
\][/tex]
[tex]\[
1 = x - 3
\][/tex]
6. Solve for [tex]\( x \)[/tex]:
Add 3 to both sides:
[tex]\[
x = 4
\][/tex]
Thus, the value of [tex]\( x \)[/tex] is [tex]\( \boxed{4} \)[/tex].
