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When deriving the Law of Cosines, the equation [tex]\(h^2 = a^2 - x^2\)[/tex] is created using the Pythagorean theorem. Use substitution to replace [tex]\(h^2\)[/tex] in the equation below and then simplify.

[tex]\[ c^2 = h^2 + b^2 - 2bx + x^2 \][/tex]

Options:
A. [tex]\(c^2 = a^2 + b^2 - 2bx\)[/tex]
B. [tex]\(c^2 = a^2 - b^2 - 2bx\)[/tex]
C. [tex]\(c^2 = a^2 - x^2 + b^2 - 2bx + x^2\)[/tex]
D. [tex]\(c^2 = h^2 + a^2 + b^2 - 2bx\)[/tex]



Answer :

GinnyAnswer
Let's start with the given equation and use substitution to simplify it step by step.

We have the equation:
[tex]\[ c^2 = h^2 + b^2 - 2bx + x^2 \][/tex]

Given that:
[tex]\[ h^2 = a^2 - x^2 \][/tex]

Step 1: Substitute [tex]\( h^2 \)[/tex] into the equation

Substitute [tex]\( h^2 = a^2 - x^2 \)[/tex] into the equation [tex]\( c^2 = h^2 + b^2 - 2bx + x^2 \)[/tex]:
[tex]\[ c^2 = (a^2 - x^2) + b^2 - 2bx + x^2 \][/tex]

Step 2: Combine like terms

Now, we will combine the like terms on the right side of the equation:
[tex]\[ c^2 = a^2 - x^2 + b^2 - 2bx + x^2 \][/tex]

Notice that [tex]\(-x^2\)[/tex] and [tex]\(x^2\)[/tex] cancel each other out:
[tex]\[ c^2 = a^2 + b^2 - 2bx \][/tex]

So, the simplified equation after substitution is:
[tex]\[ c^2 = a^2 + b^2 - 2bx \][/tex]

This is the simplified form of the equation after substituting and combining like terms.

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