SOLUTION: I need to factor 25w^2 - 121 completely. Thanks so much for your help.

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Question 260820: I need to factor 25w^2 - 121 completely.

Thanks so much for your help.

Found 2 solutions by Theo, richwmiller:
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
set 25w^2 - 121 = 0 to find the roots.

add 121 to both sides of the equation to get:

25w^2 = 121

divide both sides by 25 to get:

w^2 = 121/25

take the square root of both sides to get:

w = +/- sqrt(121)/sqrt(25)

sqrt(121) = 11
sqrt(25) = 5

equation becomes:

w = +/- 11/5

graph of this equation looks like this:

graph%28600%2C600%2C-5%2C5%2C-150%2C150%2C25%2Ax%5E2-121%29

you can see that y = 0 when x = +/- 11/5 which is equivalent to +/- 2.2

note:

to graph the equation, I substituted x for w to satisfy the graphing software requirements.


Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
Here is how to factor it
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression 25w%5E2%2B0w-121, we can see that the first coefficient is 25, the second coefficient is 0, and the last term is -121.



Now multiply the first coefficient 25 by the last term -121 to get %2825%29%28-121%29=-3025.



Now the question is: what two whole numbers multiply to -3025 (the previous product) and add to the second coefficient 0?



To find these two numbers, we need to list all of the factors of -3025 (the previous product).



Factors of -3025:

1,5,11,25,55,121,275,605,3025

-1,-5,-11,-25,-55,-121,-275,-605,-3025



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



These factors pair up and multiply to -3025.

1*(-3025) = -3025
5*(-605) = -3025
11*(-275) = -3025
25*(-121) = -3025
55*(-55) = -3025
(-1)*(3025) = -3025
(-5)*(605) = -3025
(-11)*(275) = -3025
(-25)*(121) = -3025
(-55)*(55) = -3025


Now let's add up each pair of factors to see if one pair adds to the middle coefficient 0:



First NumberSecond NumberSum
1-30251+(-3025)=-3024
5-6055+(-605)=-600
11-27511+(-275)=-264
25-12125+(-121)=-96
55-5555+(-55)=0
-13025-1+3025=3024
-5605-5+605=600
-11275-11+275=264
-25121-25+121=96
-5555-55+55=0




From the table, we can see that the two numbers -55 and 55 add to 0 (the middle coefficient).



So the two numbers -55 and 55 both multiply to -3025 and add to 0



Now replace the middle term 0w with -55w%2B55w. Remember, -55 and 55 add to 0. So this shows us that -55w%2B55w=0w.



25w%5E2%2Bhighlight%28-55w%2B55w%29-121 Replace the second term 0w with -55w%2B55w.



%2825w%5E2-55w%29%2B%2855w-121%29 Group the terms into two pairs.



5w%285w-11%29%2B%2855w-121%29 Factor out the GCF 5w from the first group.



5w%285w-11%29%2B11%285w-11%29 Factor out 11 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.



%285w%2B11%29%285w-11%29 Combine like terms. Or factor out the common term 5w-11



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



Answer:



So 25%2Aw%5E2%2B0%2Aw-121 factors to %285w%2B11%29%285w-11%29.



In other words, 25%2Aw%5E2%2B0%2Aw-121=%285w%2B11%29%285w-11%29.



Note: you can check the answer by expanding %285w%2B11%29%285w-11%29 to get 25%2Aw%5E2%2B0%2Aw-121 or by graphing the original expression and the answer (the two graphs should be identical).