SOLUTION: Factor the following: 32x^2-80x+50 Xa+ya+x+y 6xy-8x+15y-20 30x^2y+35x^2y^2 12a^2-15ab-16a+20b 4x^2+26x-48 Show work

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: Factor the following: 32x^2-80x+50 Xa+ya+x+y 6xy-8x+15y-20 30x^2y+35x^2y^2 12a^2-15ab-16a+20b 4x^2+26x-48 Show work      Log On


   

Question 640850: Factor the following:
32x^2-80x+50
Xa+ya+x+y
6xy-8x+15y-20
30x^2y+35x^2y^2
12a^2-15ab-16a+20b
4x^2+26x-48
Show work
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'll do the first one to get you started.


32x%5E2-80x%2B50 Start with the given expression.


2%2816x%5E2-40x%2B25%29 Factor out the GCF 2.


Now let's try to factor the inner expression 16x%5E2-40x%2B25


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Looking at the expression 16x%5E2-40x%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.
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.


16x%5E2%2Bhighlight%28-20x-20x%29%2B25 Replace the second term -40x with -20x-20x.


%2816x%5E2-20x%29%2B%28-20x%2B25%29 Group the terms into two pairs.


4x%284x-5%29%2B%28-20x%2B25%29 Factor out the GCF 4x from the first group.


4x%284x-5%29-5%284x-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.


%284x-5%29%284x-5%29 Combine like terms. Or factor out the common term 4x-5


%284x-5%29%5E2 Condense the terms.


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So 2%2816x%5E2-40x%2B25%29 then factors further to 2%284x-5%29%5E2


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Answer:


So 32x%5E2-80x%2B50 completely factors to 2%284x-5%29%5E2.


In other words, 32x%5E2-80x%2B50=2%284x-5%29%5E2.


Note: you can check the answer by expanding 2%284x-5%29%5E2 to get 32x%5E2-80x%2B50 or by graphing the original expression and the answer (the two graphs should be identical).


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