I will give a medal to ever answers this.
Kathy wants to factor x2 – 12x + 36 by grouping; however, Tom says it is a special product and can factor a different way. Using complete sentences explain and demonstrate how both methods will result in the same factors.

Question

I will give a medal to ever answers this.
Kathy wants to factor x2 – 12x + 36 by grouping; however, Tom says it is a special product and can factor a different way. Using complete sentences explain and demonstrate how both methods will result in the same factors.

mertsj · Answer

Kathy wants to replace -12x with -6x-6x so she will have:
x^2-6x-6x+36
She will then group the first two terms and the last two terms using the associative property to get:
(x^2-6x)+(-6x+36)
The first two terms have the common factor x.  The last two terms have the common factor -6.
Kathy will factor those out and get:
x(x-6)-6(x-6)
She now has an expression with two terms.  The two terms have the common factor x-6.
Factoring it out, Kathy has:  (x-6)(x-6).

anonymous · Answer

Thank you.

mertsj · Answer

Tom recognizes that x^2-12x+36 is of the form a^2-2ab +b^2 whose factors are (a-b)^2 where a = x and b = 6.  He replaces  a with x and b with 6 to get the factors (x-6)^2.