To solve this problem, we need to expand the expression [tex]\((b - x)^2\)[/tex]. Let's start by recalling the algebraic formula for expanding a squared binomial.
The formula for expanding [tex]\((a - b)^2\)[/tex] is given by:
[tex]\[ (a - b)^2 = a^2 - 2ab + b^2 \][/tex]
In our problem, [tex]\(a = b\)[/tex] and [tex]\(b = x\)[/tex]. So, we'll use this formula to expand [tex]\((b - x)^2\)[/tex]:
[tex]\[ (b - x)^2 = b^2 - 2bx + x^2 \][/tex]
After expanding, we get:
[tex]\[ (b - x)^2 = b^2 - 2bx + x^2 \][/tex]
We now need to identify which given option matches this expanded form when incorporated into the equation [tex]\(c^2 = h^2 + (b - x)^2\)[/tex]:
1. [tex]\(c^2 = h^2 + b^2 - x^2\)[/tex]
2. [tex]\(c^2 = h^2 + b^2 - 2bx + x^2\)[/tex]
3. [tex]\(c^2 = h^2 + b^2 + x^2\)[/tex]
4. [tex]\(c^2 = h^2 + b^2 - 2bx - x^2\)[/tex]
Substituting our expanded form [tex]\((b - x)^2 = b^2 - 2bx + x^2\)[/tex] into [tex]\(c^2 = h^2 + (b - x)^2\)[/tex]:
[tex]\[ c^2 = h^2 + b^2 - 2bx + x^2 \][/tex]
Thus, the correct matching option is:
[tex]\[ c^2 = h^2 + b^2 - 2bx + x^2 \][/tex]
So, the equation that is the result of expanding [tex]\((b - x)^2\)[/tex] is:
[tex]\[ \boxed{c^2 = h^2 + b^2 - 2bx + x^2} \][/tex]
