To find an expression equivalent to a positive value of [tex]\( x \)[/tex] given the equation [tex]\( x^2 + y^2 = 4z \)[/tex], we need to solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex] and [tex]\( z \)[/tex].
Here are the detailed steps:
1. Start with the given equation:
[tex]\[
x^2 + y^2 = 4z
\][/tex]
2. To isolate [tex]\( x^2 \)[/tex], subtract [tex]\( y^2 \)[/tex] from both sides:
[tex]\[
x^2 = 4z - y^2
\][/tex]
3. Now, take the square root of both sides to solve for [tex]\( x \)[/tex]. When taking the square root, we have two potential solutions [tex]\( x = \pm \sqrt{4z - y^2} \)[/tex]. However, because we need the expression for a positive value of [tex]\( x \)[/tex], we discard the negative solution:
[tex]\[
x = \sqrt{4z - y^2}
\][/tex]
4. Now we need to find which of the given options matches [tex]\( x = \sqrt{4z - y^2} \)[/tex].
- Option A: [tex]\( y - 2 \sqrt{z} \)[/tex] does not match because it involves subtraction and does not stem from the equation derived.
- Option B: [tex]\( 2 \sqrt{z} - y \)[/tex] also does not match for the same reasons as Option A.
- Option C: [tex]\( \sqrt{y^2 - 4z} \)[/tex] does not match as the expression inside the square root is inverted.
- Option D: [tex]\( \sqrt{4z - y^2} \)[/tex] matches exactly with the derived expression for [tex]\( x \)[/tex].
Therefore, the expression equivalent to a positive value of [tex]\( x \)[/tex] is:
[tex]\[
\boxed{\sqrt{4z - y^2}}
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{D}
\][/tex]
