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.Com
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) (Show Source): You can put this solution on YOUR website!
# 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 Number | Second Number | Sum | | 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 |
| -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,
=======================================================
# 7
Start with the given expression.
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:
Start with the difference of squares formula.
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
Answer by richard1234(7193) (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).
RELATED QUESTIONS
Factor the following expression completely:... (answered by ikleyn,MathLover1,greenestamps)
Factor the following expression completely.
x^2 - 8x + 16 - 9y^2 (answered by nyc_function,MathTherapy)
Factor the following expression completely.
x^2 - 8x + 16 - 9y^2 (answered by jim_thompson5910)
Factor completely.... (answered by chibisan)
Completely factor the following expression... (answered by sofiyac)
Completely factor the following expression:
81y^2 – 4z^2
(answered by MathLover1)
Completely factor the following expression:... (answered by Alan3354)
Factor the following expression completely:
70w^3 – 125w^2 + 30w (answered by jim_thompson5910)
Factor completely, if possible, the following expression:
{{{ 20k^2-125... (answered by jim_thompson5910)