Question

Darlene wrote this proof of the identity (x + y)2 - (x - y)2 = 4xy. Which of the following is a justification for Step 3 of her proof?
Step 1: (x + y)2 - (x - y)2 = (x + y)(x + y) - (x - y)(x - y)

Step 2: (x + y)(x + y) - (x - y)(x - y) = (x2 + xy + xy + y2) - (x2 - xy - xy + y2)

Step 3: (x2 + xy + xy + y2) - (x2 - xy - xy + y2) = (x2 + 2xy + y2) - (x2 - 2xy + y2)

Step 4: (x2 + 2xy + y2) - (x2 - 2xy + y2) = x2 + 2xy + y2 - x2 + 2xy - y2

Step 5: x2 + 2xy + y2 - x2 + 2xy - y2 = 4xy

A. Reflexive property
B. Definition of squaring a binomial
C. Combining like terms
D. Distributive property

Answer 1

Solution:

we are given that

Darlene wrote this proof of the identity (x + y)2 - (x - y)2 = 4xy.

we have been asked to find

Which of the following is a justification for Step 3 of her proof?

Step 3 of her proof is

$Step 3: (x^2 + xy + xy + y^2) - (x^2 - xy - xy + y^2) = (x^2 + 2xy + y2) - (x^2 - 2xy + y^2)$

As we can observe inside each of the paranthesis only like terms have been added. It mean the required justification is

C. Combining like terms