Let's start with the given equation:
[tex]\[ c^2 = b^2 + d^2 - 2cd \][/tex]
Our goal is to solve for [tex]\( b \)[/tex] in terms of [tex]\( c \)[/tex] and [tex]\( d \)[/tex]. Here's the step-by-step solution:
1. Rearrange the equation to isolate [tex]\( b^2 \)[/tex]:
[tex]\[ c^2 = b^2 + d^2 - 2cd \][/tex]
2. Move [tex]\( d^2 \)[/tex] and [tex]\( -2cd \)[/tex] to the other side:
[tex]\[ b^2 = c^2 - d^2 + 2cd \][/tex]
3. Take the square root of both sides to solve for [tex]\( b \)[/tex]. Since we want the positive value of [tex]\( b \)[/tex]:
[tex]\[ b = \sqrt{c^2 - d^2 + 2cd} \][/tex]
Now, let's compare this expression with the given options:
- Option A: [tex]\(\sqrt{c^2 + d^2 - 2cd}\)[/tex]
- Option B: [tex]\(\sqrt{c^2 + d^2 + 2cd}\)[/tex]
- Option C: [tex]\(\sqrt{c^2 - d^2 - 2cd}\)[/tex]
- Option D: [tex]\(\sqrt{c^2 - d^2 + 2cd}\)[/tex]
The correct expression that is equivalent to the positive value of [tex]\( b \)[/tex] is:
[tex]\[ \sqrt{c^2 - d^2 + 2cd} \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{D} \][/tex]
