chase wants to factor x^2+10x+25 by grouping; however, Paige 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.
Chase and Paige are both correct. There are at least two different ways of factoring this expression.
Grouping: To factor this expression by grouping, you will need to rewrite it, by think of 10x as 5x + 5x. (why would you do that? No great reason, except if you noticed that 25 = 5 x 5, so we will try the factors of 25.)
Rewriting the expression, we would get
x2 + 10x + 25 = x2 + (5x + 5x) + 25
Now, since all of the individual terms are connected by addition, I can rearrange the parentheses to make it easier to think of the expression as two groups.
x2 + (5x + 5x) + 25 = (x2 + 5x) + (5x + 25)
Now we want to factor each of the groups in parentheses.
(x2 + 5x) = x(x+ 5) and (5x + 25) = 5(x + 5)
So each group has one factor of (x + 5). That means we can use the distributive property in reverse, to combine the parts outside of the parentheses.
(x2 + 5x) + 5(x + 5) = x(x + 5) + 5(x + 5)
= (x + 5)(x + 5)
And that is the factored form of the expression.
Special Product: In order to recognize this expression as a special product, you need to first recognize that 25 is a perfect square equal to 5 x 5. Since that is true, you might be tempted to think about the general form of any perfect square binomial.
That means, (x + a)2 = x2 + 2ax + a2 (trying multiplying it out with FOIL and that's what you will get.) This special form says that in a perfect square binomial the constant term of the expression will be a perfect square (25 = 52), and the middle term, (the 'x'-term) will be two times the separate parts of the binomial multiplied together ( in this case 2 times x times 5 = 10x.) The expression we wanted to factor looks exactly like that: x2 + 10x + 25 = x2 + two times x times 5 + 52.
In both of these cases, you have to get used to imagining different ways of looking at the terms in the expression you are trying to factor. Patience and practice help a lot.
If you want a more visual explanation, let me know and I'll share a video with you.