SOLUTION: 6. Completely factor the following expression: 36a^2 – 84ab + 49b^2 7. Completely factor the following expression: 9y^2 – 16z^2 I do not understand please help me!I a

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: 6. Completely factor the following expression: 36a^2 – 84ab + 49b^2 7. Completely factor the following expression: 9y^2 – 16z^2 I do not understand please help me!I a      Log On


   



Question 507702: 6. Completely factor the following expression:
36a^2 – 84ab + 49b^2

7. Completely factor the following expression:
9y^2 – 16z^2

I do not understand please help me!I am desperate!!

Found 2 solutions by jim_thompson5910, richard1234:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
# 6



Looking at the expression 36a%5E2-84ab%2B49b%5E2, we can see that the first coefficient is 36, the second coefficient is -84, and the last coefficient is 49.


Now multiply the first coefficient 36 by the last coefficient 49 to get %2836%29%2849%29=1764.


Now the question is: what two whole numbers multiply to 1764 (the previous product) and add to the second coefficient -84?


To find these two numbers, we need to list all of the factors of 1764 (the previous product).


Factors of 1764:
1,2,3,4,6,7,9,12,14,18,21,28,36,42,49,63,84,98,126,147,196,252,294,441,588,882,1764
-1,-2,-3,-4,-6,-7,-9,-12,-14,-18,-21,-28,-36,-42,-49,-63,-84,-98,-126,-147,-196,-252,-294,-441,-588,-882,-1764


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


These factors pair up and multiply to 1764.
1*1764 = 1764
2*882 = 1764
3*588 = 1764
4*441 = 1764
6*294 = 1764
7*252 = 1764
9*196 = 1764
12*147 = 1764
14*126 = 1764
18*98 = 1764
21*84 = 1764
28*63 = 1764
36*49 = 1764
42*42 = 1764
(-1)*(-1764) = 1764
(-2)*(-882) = 1764
(-3)*(-588) = 1764
(-4)*(-441) = 1764
(-6)*(-294) = 1764
(-7)*(-252) = 1764
(-9)*(-196) = 1764
(-12)*(-147) = 1764
(-14)*(-126) = 1764
(-18)*(-98) = 1764
(-21)*(-84) = 1764
(-28)*(-63) = 1764
(-36)*(-49) = 1764
(-42)*(-42) = 1764

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -84:


First NumberSecond NumberSum
117641+1764=1765
28822+882=884
35883+588=591
44414+441=445
62946+294=300
72527+252=259
91969+196=205
1214712+147=159
1412614+126=140
189818+98=116
218421+84=105
286328+63=91
364936+49=85
424242+42=84
-1-1764-1+(-1764)=-1765
-2-882-2+(-882)=-884
-3-588-3+(-588)=-591
-4-441-4+(-441)=-445
-6-294-6+(-294)=-300
-7-252-7+(-252)=-259
-9-196-9+(-196)=-205
-12-147-12+(-147)=-159
-14-126-14+(-126)=-140
-18-98-18+(-98)=-116
-21-84-21+(-84)=-105
-28-63-28+(-63)=-91
-36-49-36+(-49)=-85
-42-42-42+(-42)=-84



From the table, we can see that the two numbers -42 and -42 add to -84 (the middle coefficient).


So the two numbers -42 and -42 both multiply to 1764 and add to -84


Now replace the middle term -84ab with -42ab-42ab. Remember, -42 and -42 add to -84. So this shows us that -42ab-42ab=-84ab.


36a%5E2%2Bhighlight%28-42ab-42ab%29%2B49b%5E2 Replace the second term -84ab with -42ab-42ab.


%2836a%5E2-42ab%29%2B%28-42ab%2B49b%5E2%29 Group the terms into two pairs.


6a%286a-7b%29%2B%28-42ab%2B49b%5E2%29 Factor out the GCF 6a from the first group.


6a%286a-7b%29-7b%286a-7b%29 Factor out -7b from the second group.


%286a-7b%29%286a-7b%29 Factor out the GCF 6a-7b


%286a-7b%29%5E2 Condense the terms.


So 36a%5E2-84ab%2B49b%5E2 completely factors to %286a-7b%29%5E2


In other words, 36a%5E2-84ab%2B49b%5E2=%286a-7b%29%5E2
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# 7


9y%5E2-16z%5E2 Start with the given expression.


%283y%29%5E2-16z%5E2 Rewrite 9y%5E2 as %283y%29%5E2.


%283y%29%5E2-%284z%29%5E2 Rewrite 16z%5E2 as %284z%29%5E2.


Notice how we have a difference of squares A%5E2-B%5E2 where in this case A=3y and B=4z.


So let's use the difference of squares formula A%5E2-B%5E2=%28A%2BB%29%28A-B%29 to factor the expression:


A%5E2-B%5E2=%28A%2BB%29%28A-B%29 Start with the difference of squares formula.


%283y%29%5E2-%284z%29%5E2=%283y%2B4z%29%283y-4z%29 Plug in A=3y and B=4z.


So this shows us that 9y%5E2-16z%5E2 factors to %283y%2B4z%29%283y-4z%29.


In other words 9y%5E2-16z%5E2=%283y%2B4z%29%283y-4z%29.


Let me know if you need more help or if you need me to explain a step in more detail.
Feel free to email me at jim_thompson5910@hotmail.com
or you can visit my website here: http://www.freewebs.com/jimthompson5910/home.html

Thanks,

Jim

Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
6. Noting that 36 = 6^2, 49 = 7^2, and 84 = 6*7, intuitively we would write something like

(6a - 7b)^2 (note the minus because the ab coefficient is -84 instead of 84).

Writing the factored form out of nowhere and no trial-and-error takes some practice and experience.

7. This is a difference of two squares, and can be written as (3y)^2 - (4z)^2, or (3y + 4z)(3y - 4z).