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In the proof of the Law of Cosines, the equation [tex]c^2 = h^2 + (b-x)^2[/tex] was created using the Pythagorean theorem. Which equation is a result of expanding [tex](b-x)^2[/tex]?
A. [tex]c^2 = h^2 + b^2 - x^2[/tex] B. [tex]c^2 = h^2 + b^2 - 2bx + x^2[/tex] C. [tex]c^2 = h^2 + b^2 + x^2[/tex] D. [tex]c^2 = h^2 + b^2 - 2bx - x^2[/tex]
To solve this problem, we need to correctly apply the algebraic expansion of \((b - x)^2\) and then substitute it into the given equation \(c^2 = h^2 + (b - x)^2\).
### Step-by-Step Solution:
1. Start with the expression \((b - x)^2\): [tex]\[(b - x)^2\][/tex]
2. Expand this expression using the formula \((a - b)^2 = a^2 - 2ab + b^2\) where \(a = b\) and \(b = x\): [tex]\[
(b - x)^2 = b^2 - 2bx + x^2
\][/tex]
3. Substitute the expanded form of \((b - x)^2\) into the original equation \(c^2 = h^2 + (b - x)^2\): [tex]\[
c^2 = h^2 + (b^2 - 2bx + x^2)
\][/tex]
This is the expanded form of \((b - x)^2\) in the given equation.
### Conclusion: The correct equation resulting from expanding \((b - x)^2\) and substituting it back into \(c^2 = h^2 + (b - x)^2\) is: [tex]\[
c^2 = h^2 + b^2 - 2bx + x^2
\][/tex]
Thus, the correct answer is: [tex]\[
\boxed{c^2 = h^2 + b^2 - 2 b x + x^2}
\][/tex]
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