Question

Find the two products below. Compare and contrast, in complete sentences, the similarities and differences of the two.

(x + 6)(x - 6) and (x - 6)(x - 6)

Answer 1

(a+b)(a-b) = a^2 - b^2
(a-b)(a-b) = (a-b)^2 = a^2 - 2ab - b^2

Answer 2

Definitions:
A NUMBER represents a specific, measurable, VALUE.
A VARIABLE represents an UNKNOWN value, often represented by a LETTER within an equation. It is usually the goal to discover the VALUE of this UNKNOWN.
This is an example of a critical method in mathematics, the process of DISTRIBUTED MULTIPLICATION.
Since you have a group of terms, related, or connected by addition within parentheses, and they are put together, this means multiply the contents of the parentheses to each other. Any time you have this situation the process is the same.
(x+6) (x-6)
Take the first term of the first parentheses. In this case, x. Then you multiply x by the first term in the second set of parentheses, in this case, x. Then you take x, the first term of the first set of parentheses, again, and multiply it with the next term of the second set of parentheses, in this case, -6.
You should now have written the first step of the answer, x times x is x2, and x times -6 is -6x…. OR
X2 – 6X

Now you move on, smoothly, to the second TERM of the FIRST pair of parentheses, or in this case, 6. This term also will be MULTIPLIED to the FIRST term and then to the SECOND term of the SECOND set of parentheses. In other words, 6 times X is 6X, and 6 times -6 is -36. This is now what should be written.
X2 – 6X +6X and -36.
Now you would combine LIKE terms. X2 -36. NOTICE that since BOTH parentheses contain the SAME terms with the only difference being the CONNECTING sign that in the process of DISTRIBUTIVE MULTIPLICATION, the terms that MIX the NUMBER with the VARIABLE CANCEL each other out.
Now you do the same ROUTINE, it is ALWAYS a ROUTINE, do the SAME thing with next set of NUMBERS, and TERMS.
(x-6)(x-6)
You should get X2 – 12X -36.
If you don’t, go back and work out these again now. You CAN and MUST understand DISTRIBUTIVE MULTIPLICATION.
The two answers are SIMILAR in that the far end TERMS, which also happen to be the terms that result from multiplying VARIABLE with VARIABLE, and NUMBER with NUMBER, LIKE with LIKE, are the SAME. It is the middle TERMS, those TERMS that are affected by the SIGN that CONNECTS one set of the pair within the parentheses.