Which expression is missing from step 7?

A. [tex]\(-2 d e\)[/tex]
B. [tex]\((d-e)^2\)[/tex]
C. [tex]\(A^2 + B^2\)[/tex]
D. [tex]\((A + B)^2\)[/tex]



Answer :

To find the missing expression in step 7, let's carefully analyze the steps provided.

1. Start with the given equation from the problem:
[tex]\[ \left(\sqrt{1+d^2}\right)^2 + \left(\sqrt{e^2+1}\right)^2 \][/tex]
By squaring the expressions inside the square roots, we get:
[tex]\[ (1 + d^2) + (e^2 + 1) = d^2 - 2de + e^2 \][/tex]

2. Rewrite the equation:
[tex]\[ (1 + d^2) + (e^2 + 1) = d^2 - 2de + e^2 \][/tex]

3. Simplify the left side:
[tex]\[ 2 + d^2 + e^2 = d^2 - 2de + e^2 \][/tex]

4. As per step 7, the simplified form is:
[tex]\[ 2 = -2de \][/tex]

By comparing both sides of the equation [tex]\( 2 = -2de \)[/tex], the only way to achieve simplification steps by removing common terms and maintaining equality is if:
[tex]\[ d^2 + e^2 + 2 = d^2 - 2de + e^2 \implies 2 = -2de \][/tex]

Given this simplification, it is evident that the term required to set up the equation is:
[tex]\[ (d-e)^2 \][/tex]

Therefore, the missing expression from step 7 is:
[tex]\[ \boxed{(d-e)^2} \][/tex]

Thus, the correct answer is:
[tex]\[ B. (d-e)^2 \][/tex]