# 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 ->  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

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

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

Found 2 solutions by jim_thompson5910, richard1234:
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# 6

Looking at the expression , we can see that the first coefficient is , the second coefficient is , and the last coefficient is .

Now multiply the first coefficient by the last coefficient to get .

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

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

Factors of :
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 .
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 :

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 and add to (the middle coefficient).

So the two numbers and both multiply to and add to

Now replace the middle term with . Remember, and add to . So this shows us that .

Replace the second term with .

Group the terms into two pairs.

Factor out the GCF from the first group.

Factor out from the second group.

Factor out the GCF

Condense the terms.

So completely factors to

In other words,
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# 7

Rewrite as .

Rewrite as .

Notice how we have a difference of squares where in this case and .

So let's use the difference of squares formula to factor the expression:

Plug in and .

So this shows us that factors to .

In other words .

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