SOLUTION: factor the quadratic expression completely, and find the roots of the expression. 66x2 - 130x + 56

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Question 601653: factor the quadratic expression completely, and find the roots of the expression.
66x2 - 130x + 56
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

66x%5E2-130x%2B56 Start with the given expression.


2%2833x%5E2-65x%2B28%29 Factor out the GCF 2.


Now let's try to factor the inner expression 33x%5E2-65x%2B28


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Looking at the expression 33x%5E2-65x%2B28, we can see that the first coefficient is 33, the second coefficient is -65, and the last term is 28.


Now multiply the first coefficient 33 by the last term 28 to get %2833%29%2828%29=924.


Now the question is: what two whole numbers multiply to 924 (the previous product) and add to the second coefficient -65?


To find these two numbers, we need to list all of the factors of 924 (the previous product).


Factors of 924:
1,2,3,4,6,7,11,12,14,21,22,28,33,42,44,66,77,84,132,154,231,308,462,924
-1,-2,-3,-4,-6,-7,-11,-12,-14,-21,-22,-28,-33,-42,-44,-66,-77,-84,-132,-154,-231,-308,-462,-924


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


These factors pair up and multiply to 924.
1*924 = 924
2*462 = 924
3*308 = 924
4*231 = 924
6*154 = 924
7*132 = 924
11*84 = 924
12*77 = 924
14*66 = 924
21*44 = 924
22*42 = 924
28*33 = 924
(-1)*(-924) = 924
(-2)*(-462) = 924
(-3)*(-308) = 924
(-4)*(-231) = 924
(-6)*(-154) = 924
(-7)*(-132) = 924
(-11)*(-84) = 924
(-12)*(-77) = 924
(-14)*(-66) = 924
(-21)*(-44) = 924
(-22)*(-42) = 924
(-28)*(-33) = 924

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -65:


First NumberSecond NumberSum
19241+924=925
24622+462=464
33083+308=311
42314+231=235
61546+154=160
71327+132=139
118411+84=95
127712+77=89
146614+66=80
214421+44=65
224222+42=64
283328+33=61
-1-924-1+(-924)=-925
-2-462-2+(-462)=-464
-3-308-3+(-308)=-311
-4-231-4+(-231)=-235
-6-154-6+(-154)=-160
-7-132-7+(-132)=-139
-11-84-11+(-84)=-95
-12-77-12+(-77)=-89
-14-66-14+(-66)=-80
-21-44-21+(-44)=-65
-22-42-22+(-42)=-64
-28-33-28+(-33)=-61



From the table, we can see that the two numbers -21 and -44 add to -65 (the middle coefficient).


So the two numbers -21 and -44 both multiply to 924 and add to -65


Now replace the middle term -65x with -21x-44x. Remember, -21 and -44 add to -65. So this shows us that -21x-44x=-65x.


33x%5E2%2Bhighlight%28-21x-44x%29%2B28 Replace the second term -65x with -21x-44x.


%2833x%5E2-21x%29%2B%28-44x%2B28%29 Group the terms into two pairs.


3x%2811x-7%29%2B%28-44x%2B28%29 Factor out the GCF 3x from the first group.


3x%2811x-7%29-4%2811x-7%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.


%283x-4%29%2811x-7%29 Combine like terms. Or factor out the common term 11x-7


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So 2%2833x%5E2-65x%2B28%29 then factors further to 2%283x-4%29%2811x-7%29


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Answer:


So 66x%5E2-130x%2B56 completely factors to 2%283x-4%29%2811x-7%29.


In other words, 66x%5E2-130x%2B56=2%283x-4%29%2811x-7%29.


Note: you can check the answer by expanding 2%283x-4%29%2811x-7%29 to get 66x%5E2-130x%2B56 or by graphing the original expression and the answer (the two graphs should be identical).

I'll leave finding the roots to you.