SOLUTION: Use a special product formula to factor the perfect square trinomial. 25x^2-40x+16

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Question 110949: Use a special product formula to factor the perfect square trinomial.

25x^2-40x+16
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression 25x%5E2-40x%2B16, we can see that the first coefficient is 25, the second coefficient is -40, and the last term is 16.



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

1*400 = 400
2*200 = 400
4*100 = 400
5*80 = 400
8*50 = 400
10*40 = 400
16*25 = 400
20*20 = 400
(-1)*(-400) = 400
(-2)*(-200) = 400
(-4)*(-100) = 400
(-5)*(-80) = 400
(-8)*(-50) = 400
(-10)*(-40) = 400
(-16)*(-25) = 400
(-20)*(-20) = 400


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-20x. Remember, -20 and -20 add to -40. So this shows us that -20x-20x=-40x.



25x%5E2%2Bhighlight%28-20x-20x%29%2B16 Replace the second term -40x with -20x-20x.



%2825x%5E2-20x%29%2B%28-20x%2B16%29 Group the terms into two pairs.



5x%285x-4%29%2B%28-20x%2B16%29 Factor out the GCF 5x from the first group.



5x%285x-4%29-4%285x-4%29 Factor out 4 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.



%285x-4%29%285x-4%29 Combine like terms. Or factor out the common term 5x-4



%285x-4%29%5E2 Condense the terms.



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Answer:



So 25%2Ax%5E2-40%2Ax%2B16 factors to %285x-4%29%5E2.



In other words, 25%2Ax%5E2-40%2Ax%2B16=%285x-4%29%5E2.



Note: you can check the answer by expanding %285x-4%29%5E2 to get 25%2Ax%5E2-40%2Ax%2B16 or by graphing the original expression and the answer (the two graphs should be identical).