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]
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]
