SOLUTION: 16x^2 + 40x + 25

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Question 143271: 16x^2 + 40x + 25
Answer by jim_thompson5910(35256) About Me  (Show Source):
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Looking at 16x%5E2%2B40x%2B25 we can see that the first term is 16x%5E2 and the last term is 25 where the coefficients are 16 and 25 respectively.

Now multiply the first coefficient 16 and the last coefficient 25 to get 400. Now what two numbers multiply to 400 and add to the middle coefficient 40? Let's list all of the factors of 400:



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

-1,-2,-4,-5,-8,-10,-16,-20,-25,-40,-50,-80,-100,-200 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 400
1*400
2*200
4*100
5*80
8*50
10*40
16*25
20*20
(-1)*(-400)
(-2)*(-200)
(-4)*(-100)
(-5)*(-80)
(-8)*(-50)
(-10)*(-40)
(-16)*(-25)
(-20)*(-20)

note: remember two negative numbers multiplied together make a positive number


Now which of these pairs add to 40? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 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 this list we can see that 20 and 20 add up to 40 and multiply to 400


Now looking at the expression 16x%5E2%2B40x%2B25, replace 40x with 20x%2B20x (notice 20x%2B20x adds up to 40x. So it is equivalent to 40x)

16x%5E2%2Bhighlight%2820x%2B20x%29%2B25


Now let's factor 16x%5E2%2B20x%2B20x%2B25 by grouping:


%2816x%5E2%2B20x%29%2B%2820x%2B25%29 Group like terms


4x%284x%2B5%29%2B5%284x%2B5%29 Factor out the GCF of 4x out of the first group. Factor out the GCF of 5 out of the second group


%284x%2B5%29%284x%2B5%29 Since we have a common term of 4x%2B5, we can combine like terms

So 16x%5E2%2B20x%2B20x%2B25 factors to %284x%2B5%29%284x%2B5%29


So this also means that 16x%5E2%2B40x%2B25 factors to %284x%2B5%29%284x%2B5%29 (since 16x%5E2%2B40x%2B25 is equivalent to 16x%5E2%2B20x%2B20x%2B25)


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