SOLUTION: Could you help me with this problem about factoring polynomials completely 4x^2-20x+25 Thank you for your time and help!

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Question 640408: Could you help me with this problem about factoring polynomials completely
4x^2-20x+25
Thank you for your time and help!

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression 4x%5E2-20x%2B25, we can see that the first coefficient is 4, the second coefficient is -20, and the last term is 25.


Now multiply the first coefficient 4 by the last term 25 to get %284%29%2825%29=100.


Now the question is: what two whole numbers multiply to 100 (the previous product) and add to the second coefficient -20?


To find these two numbers, we need to list all of the factors of 100 (the previous product).


Factors of 100:
1,2,4,5,10,20,25,50,100
-1,-2,-4,-5,-10,-20,-25,-50,-100


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


These factors pair up and multiply to 100.
1*100 = 100
2*50 = 100
4*25 = 100
5*20 = 100
10*10 = 100
(-1)*(-100) = 100
(-2)*(-50) = 100
(-4)*(-25) = 100
(-5)*(-20) = 100
(-10)*(-10) = 100

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -20:


First NumberSecond NumberSum
11001+100=101
2502+50=52
4254+25=29
5205+20=25
101010+10=20
-1-100-1+(-100)=-101
-2-50-2+(-50)=-52
-4-25-4+(-25)=-29
-5-20-5+(-20)=-25
-10-10-10+(-10)=-20



From the table, we can see that the two numbers -10 and -10 add to -20 (the middle coefficient).


So the two numbers -10 and -10 both multiply to 100 and add to -20


Now replace the middle term -20x with -10x-10x. Remember, -10 and -10 add to -20. So this shows us that -10x-10x=-20x.


4x%5E2%2Bhighlight%28-10x-10x%29%2B25 Replace the second term -20x with -10x-10x.


%284x%5E2-10x%29%2B%28-10x%2B25%29 Group the terms into two pairs.


2x%282x-5%29%2B%28-10x%2B25%29 Factor out the GCF 2x from the first group.


2x%282x-5%29-5%282x-5%29 Factor out 5 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.


%282x-5%29%282x-5%29 Combine like terms. Or factor out the common term 2x-5


%282x-5%29%5E2 Condense the terms.


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Answer:


So 4x%5E2-20x%2B25 factors to %282x-5%29%5E2.


In other words, 4x%5E2-20x%2B25=%282x-5%29%5E2.


Note: you can check the answer by expanding %282x-5%29%5E2 to get 4x%5E2-20x%2B25 or by graphing the original expression and the answer (the two graphs should be identical).