SOLUTION: I am having the worst time trying to solve this one. Here is what I have: 25x^2 + 10x + 1 I come up with (5x + 2)(5x ) and then i am lost! Please help!!

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Question 267280: I am having the worst time trying to solve this one. Here is what I have:
25x^2 + 10x + 1
I come up with (5x + 2)(5x )
and then i am lost!
Please help!!
Found 3 solutions by jim_thompson5910, stanbon, solver91311:
Answer by jim_thompson5910(35256)   (Show Source): You can put this solution on YOUR website!
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression , we can see that the first coefficient is , the second coefficient is , and the last term is .



Now multiply the first coefficient by the last term to get .



Now the question is: what two whole numbers multiply to (the previous product) and add to the second coefficient ?



To find these two numbers, we need to list all of the factors of (the previous product).



Factors of :

1,5,25

-1,-5,-25



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to .

1*25 = 25
5*5 = 25
(-1)*(-25) = 25
(-5)*(-5) = 25


Now let's add up each pair of factors to see if one pair adds to the middle coefficient :



First NumberSecond NumberSum
1251+25=26
555+5=10
-1-25-1+(-25)=-26
-5-5-5+(-5)=-10




From the table, we can see that the two numbers and add to (the middle coefficient).



So the two numbers and both multiply to and add to



Now replace the middle term with . Remember, and add to . So this shows us that .



Replace the second term with .



Group the terms into two pairs.



Factor out the GCF from the first group.



Factor out from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.



Combine like terms. Or factor out the common term



Condense the terms.



===============================================================



Answer:



So factors to .



In other words, .



Note: you can check the answer by expanding to get or by graphing the original expression and the answer (the two graphs should be identical).
Answer by stanbon(75887)   (Show Source): You can put this solution on YOUR website!
25x^2 + 10x + 1
-----------------------
Notice that the 1st and the 3rd terms are perfect squares
---
The square roots are 5x and 1
----
The 2nd term is 2 times the product of those square roots: 2*5x*1 = 10x
--------------------------------------------
NOTE: A trinomial that meets those conditions is called
a Perfect trinomial square.
----
Factoring a perfect trinomial square:
[(squre rt. of 1st term) sign of the 2nd term (square rt. of the 3rd term)^2
Your Problem:
(5x+1)^2
or you can write it as (5x+1)(5x+1)
======================================
Cheers,
Stan H.
======================================
Answer by solver91311(24713)   (Show Source): You can put this solution on YOUR website!


First let's check to see if your factorization is correct. We know that if we say that and are the factors of , then we must be able to multiply times and get as a result. Let's check your factors:



But you started with and we can say for certain that (At least until someone invents a mathematics where 1 equals 0).

No wonder you are lost.

Let's see: 5 times 5 is 25, 1 times 1 is 1, and 5 plus 5 is 10. So how about:



Let's check:

First

Outside

Inside

Last

Putting it all together:



Collect like terms:



Yep, that checks. Your factors are indeed:



Which can also be expressed:




John
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