SOLUTION: factor completely 25m^2+81-90m

Algebra ->  Distributive-associative-commutative-properties -> SOLUTION: factor completely 25m^2+81-90m      Log On


   

Question 201025: factor completely
25m^2+81-90m
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
25m%5E2%2B81-90m Start with the given expression.


25m%5E2-90m%2B81 Rearrange the terms in descending exponential order.



Looking at the expression 25m%5E2-90m%2B81, we can see that the first coefficient is 25, the second coefficient is -90, and the last term is 81.


Now multiply the first coefficient 25 by the last term 81 to get %2825%29%2881%29=2025.


Now the question is: what two whole numbers multiply to 2025 (the previous product) and add to the second coefficient -90?


To find these two numbers, we need to list all of the factors of 2025 (the previous product).


Factors of 2025:
1,3,5,9,15,25,27,45,75,81,135,225,405,675,2025
-1,-3,-5,-9,-15,-25,-27,-45,-75,-81,-135,-225,-405,-675,-2025


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


These factors pair up and multiply to 2025.
1*2025
3*675
5*405
9*225
15*135
25*81
27*75
45*45
(-1)*(-2025)
(-3)*(-675)
(-5)*(-405)
(-9)*(-225)
(-15)*(-135)
(-25)*(-81)
(-27)*(-75)
(-45)*(-45)

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -90:


First NumberSecond NumberSum
120251+2025=2026
36753+675=678
54055+405=410
92259+225=234
1513515+135=150
258125+81=106
277527+75=102
454545+45=90
-1-2025-1+(-2025)=-2026
-3-675-3+(-675)=-678
-5-405-5+(-405)=-410
-9-225-9+(-225)=-234
-15-135-15+(-135)=-150
-25-81-25+(-81)=-106
-27-75-27+(-75)=-102
-45-45-45+(-45)=-90



From the table, we can see that the two numbers -45 and -45 add to -90 (the middle coefficient).


So the two numbers -45 and -45 both multiply to 2025 and add to -90


Now replace the middle term -90m with -45m-45m. Remember, -45 and -45 add to -90. So this shows us that -45m-45m=-90m.


25m%5E2%2Bhighlight%28-45m-45m%29%2B81 Replace the second term -90m with -45m-45m.


%2825m%5E2-45m%29%2B%28-45m%2B81%29 Group the terms into two pairs.


5m%285m-9%29%2B%28-45m%2B81%29 Factor out the GCF 5m from the first group.


5m%285m-9%29-9%285m-9%29 Factor out 9 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.


%285m-9%29%285m-9%29 Combine like terms. Or factor out the common term 5m-9


%285m-9%29%5E2 Condense

---------------------------------------------


Answer:


So 25m%5E2%2B81-90m factors to %285m-9%29%5E2.


In other words, 25m%5E2%2B81-90m=%285m-9%29%5E2


Note: you can check the answer by FOILing %285m-9%29%5E2 to get 25m%5E2%2B81-90m or by graphing the original expression and the answer (the two graphs should be identical).