SOLUTION: 20y^2-25y+5

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Question 596104: 20y^2-25y+5
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'm assuming you want to factor.



20y%5E2-25y%2B5 Start with the given expression.


5%284y%5E2-5y%2B1%29 Factor out the GCF 5.


Now let's try to factor the inner expression 4y%5E2-5y%2B1


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Looking at the expression 4y%5E2-5y%2B1, we can see that the first coefficient is 4, the second coefficient is -5, and the last term is 1.


Now multiply the first coefficient 4 by the last term 1 to get %284%29%281%29=4.


Now the question is: what two whole numbers multiply to 4 (the previous product) and add to the second coefficient -5?


To find these two numbers, we need to list all of the factors of 4 (the previous product).


Factors of 4:
1,2,4
-1,-2,-4


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


These factors pair up and multiply to 4.
1*4 = 4
2*2 = 4
(-1)*(-4) = 4
(-2)*(-2) = 4

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -5:


First NumberSecond NumberSum
141+4=5
222+2=4
-1-4-1+(-4)=-5
-2-2-2+(-2)=-4



From the table, we can see that the two numbers -1 and -4 add to -5 (the middle coefficient).


So the two numbers -1 and -4 both multiply to 4 and add to -5


Now replace the middle term -5y with -y-4y. Remember, -1 and -4 add to -5. So this shows us that -y-4y=-5y.


4y%5E2%2Bhighlight%28-y-4y%29%2B1 Replace the second term -5y with -y-4y.


%284y%5E2-y%29%2B%28-4y%2B1%29 Group the terms into two pairs.


y%284y-1%29%2B%28-4y%2B1%29 Factor out the GCF y from the first group.


y%284y-1%29-1%284y-1%29 Factor out 1 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.


%28y-1%29%284y-1%29 Combine like terms. Or factor out the common term 4y-1


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So 5%284y%5E2-5y%2B1%29 then factors further to 5%28y-1%29%284y-1%29


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Answer:


So 20y%5E2-25y%2B5 completely factors to 5%28y-1%29%284y-1%29.


In other words, 20y%5E2-25y%2B5=5%28y-1%29%284y-1%29.


Note: you can check the answer by expanding 5%28y-1%29%284y-1%29 to get 20y%5E2-25y%2B5 or by graphing the original expression and the answer (the two graphs should be identical).