SOLUTION: 16x2 + 40x + 25 Factor competely

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Question 148771: 16x2 + 40x + 25 Factor competely
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression 16x%5E2%2B40x%2B25, we can see that the first coefficient is 16, the second coefficient is 40, and the last term is 25.


Now multiply the first coefficient 16 by the last term 25 to get %2816%29%2825%29=400.


Now the question is: what two whole numbers multiply to 400 (the previous product) and add to the second coefficient 40?


To find these two numbers, we need to list all of the factors of 400 (the previous product).


Factors of 400:
1,2,4,5,8,10,16,20,25,40,50,80,100,200,400
-1,-2,-4,-5,-8,-10,-16,-20,-25,-40,-50,-80,-100,-200,-400


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


These factors pair up and multiply to 400. For instance, 1%2A400=400, 2%2A200=400, etc.


Since 400 is positive, this means that either
a) both factors are positive, or...
b) both factors are negative.


Now let's add up each pair of factors to see if one pair adds to the middle coefficient 40:


First NumberSecond NumberSum
14001+400=401
22002+200=202
41004+100=104
5805+80=85
8508+50=58
104010+40=50
162516+25=41
202020+20=40
-1-400-1+(-400)=-401
-2-200-2+(-200)=-202
-4-100-4+(-100)=-104
-5-80-5+(-80)=-85
-8-50-8+(-50)=-58
-10-40-10+(-40)=-50
-16-25-16+(-25)=-41
-20-20-20+(-20)=-40



From the table, we can see that the two numbers 20 and 20 add to 40 (the middle coefficient).


So the two numbers 20 and 20 both multiply to 400 and add to 40


Now replace the middle term 40x with 20x%2B20x. Remember, 20 and 20 add to 40. So this shows us that 20x%2B20x=40x.


16x%5E2%2Bhighlight%2820x%2B20x%29%2B25 Replace the second term 40x with 20x%2B20x.


%2816x%5E2%2B20x%29%2B%2820x%2B25%29 Group the terms into two pairs.


4x%284x%2B5%29%2B%2820x%2B25%29 Factor out the GCF 4x from the first group.


4x%284x%2B5%29%2B5%284x%2B5%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.


%284x%2B5%29%284x%2B5%29 Combine like terms. Or factor out the common term 4x%2B5



note: %284x%2B5%29%284x%2B5%29 is equivalent to %284x%2B5%29%5E2 since the term 4x%2B5 occurs twice. So 16x%5E2%2B40x%2B25 also factors to %284x%2B5%29%5E2



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Answer:


So 16x%5E2%2B40x%2B25 factors to %284x%2B5%29%5E2.


Note: you can check the answer by FOILing %284x%2B5%29%5E2 to get 16x%5E2%2B40x%2B25 or by graphing the original expression and the answer (the two graphs should be identical).