SOLUTION: please help me solve this question {{{64x^2-16x+1}}} please explain as you solve the problem thank you

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Question 321094: please help me solve this question 64x%5E2-16x%2B1
please explain as you solve the problem
thank you
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'm assuming you want to factor.


Looking at the expression 64x%5E2-16x%2B1, we can see that the first coefficient is 64, the second coefficient is -16, and the last term is 1.


Now multiply the first coefficient 64 by the last term 1 to get %2864%29%281%29=64.


Now the question is: what two whole numbers multiply to 64 (the previous product) and add to the second coefficient -16?


To find these two numbers, we need to list all of the factors of 64 (the previous product).


Factors of 64:
1,2,4,8,16,32,64
-1,-2,-4,-8,-16,-32,-64


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


These factors pair up and multiply to 64.
1*64 = 64
2*32 = 64
4*16 = 64
8*8 = 64
(-1)*(-64) = 64
(-2)*(-32) = 64
(-4)*(-16) = 64
(-8)*(-8) = 64

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -16:


First NumberSecond NumberSum
1641+64=65
2322+32=34
4164+16=20
888+8=16
-1-64-1+(-64)=-65
-2-32-2+(-32)=-34
-4-16-4+(-16)=-20
-8-8-8+(-8)=-16



From the table, we can see that the two numbers -8 and -8 add to -16 (the middle coefficient).


So the two numbers -8 and -8 both multiply to 64 and add to -16


Now replace the middle term -16x with -8x-8x. Remember, -8 and -8 add to -16. So this shows us that -8x-8x=-16x.


64x%5E2%2Bhighlight%28-8x-8x%29%2B1 Replace the second term -16x with -8x-8x.


%2864x%5E2-8x%29%2B%28-8x%2B1%29 Group the terms into two pairs.


8x%288x-1%29%2B%28-8x%2B1%29 Factor out the GCF 8x from the first group.


8x%288x-1%29-1%288x-1%29 Factor out 1 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.


%288x-1%29%288x-1%29 Combine like terms. Or factor out the common term 8x-1


%288x-1%29%5E2 Condense the terms.


===============================================================


Answer:


So 64x%5E2-16x%2B1 factors to %288x-1%29%5E2.


In other words, 64x%5E2-16x%2B1=%288x-1%29%5E2.


Note: you can check the answer by expanding %288x-1%29%5E2 to get 64x%5E2-16x%2B1 or by graphing the original expression and the answer (the two graphs should be identical).