SOLUTION: 35x^2+3xy-54y^2

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: 35x^2+3xy-54y^2      Log On


   

Question 640704: 35x^2+3xy-54y^2
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'm assuming you want to factor this.



Looking at the expression 35x%5E2%2B3xy-54y%5E2, we can see that the first coefficient is 35, the second coefficient is 3, and the last coefficient is -54.


Now multiply the first coefficient 35 by the last coefficient -54 to get %2835%29%28-54%29=-1890.


Now the question is: what two whole numbers multiply to -1890 (the previous product) and add to the second coefficient 3?


To find these two numbers, we need to list all of the factors of -1890 (the previous product).


Factors of -1890:
1,2,3,5,6,7,9,10,14,15,18,21,27,30,35,42,45,54,63,70,90,105,126,135,189,210,270,315,378,630,945,1890
-1,-2,-3,-5,-6,-7,-9,-10,-14,-15,-18,-21,-27,-30,-35,-42,-45,-54,-63,-70,-90,-105,-126,-135,-189,-210,-270,-315,-378,-630,-945,-1890


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


These factors pair up and multiply to -1890.
1*(-1890) = -1890
2*(-945) = -1890
3*(-630) = -1890
5*(-378) = -1890
6*(-315) = -1890
7*(-270) = -1890
9*(-210) = -1890
10*(-189) = -1890
14*(-135) = -1890
15*(-126) = -1890
18*(-105) = -1890
21*(-90) = -1890
27*(-70) = -1890
30*(-63) = -1890
35*(-54) = -1890
42*(-45) = -1890
(-1)*(1890) = -1890
(-2)*(945) = -1890
(-3)*(630) = -1890
(-5)*(378) = -1890
(-6)*(315) = -1890
(-7)*(270) = -1890
(-9)*(210) = -1890
(-10)*(189) = -1890
(-14)*(135) = -1890
(-15)*(126) = -1890
(-18)*(105) = -1890
(-21)*(90) = -1890
(-27)*(70) = -1890
(-30)*(63) = -1890
(-35)*(54) = -1890
(-42)*(45) = -1890

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 3:


First NumberSecond NumberSum
1-18901+(-1890)=-1889
2-9452+(-945)=-943
3-6303+(-630)=-627
5-3785+(-378)=-373
6-3156+(-315)=-309
7-2707+(-270)=-263
9-2109+(-210)=-201
10-18910+(-189)=-179
14-13514+(-135)=-121
15-12615+(-126)=-111
18-10518+(-105)=-87
21-9021+(-90)=-69
27-7027+(-70)=-43
30-6330+(-63)=-33
35-5435+(-54)=-19
42-4542+(-45)=-3
-11890-1+1890=1889
-2945-2+945=943
-3630-3+630=627
-5378-5+378=373
-6315-6+315=309
-7270-7+270=263
-9210-9+210=201
-10189-10+189=179
-14135-14+135=121
-15126-15+126=111
-18105-18+105=87
-2190-21+90=69
-2770-27+70=43
-3063-30+63=33
-3554-35+54=19
-4245-42+45=3



From the table, we can see that the two numbers -42 and 45 add to 3 (the middle coefficient).


So the two numbers -42 and 45 both multiply to -1890 and add to 3


Now replace the middle term 3xy with -42xy%2B45xy. Remember, -42 and 45 add to 3. So this shows us that -42xy%2B45xy=3xy.


35x%5E2%2Bhighlight%28-42xy%2B45xy%29-54y%5E2 Replace the second term 3xy with -42xy%2B45xy.


%2835x%5E2-42xy%29%2B%2845xy-54y%5E2%29 Group the terms into two pairs.


7x%285x-6y%29%2B%2845xy-54y%5E2%29 Factor out the GCF 7x from the first group.


7x%285x-6y%29%2B9y%285x-6y%29 Factor out 9y 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.


%287x%2B9y%29%285x-6y%29 Combine like terms. Or factor out the common term 5x-6y


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


Answer:


So 35x%5E2%2B3xy-54y%5E2 factors to %287x%2B9y%29%285x-6y%29.


In other words, 35x%5E2%2B3xy-54y%5E2=%287x%2B9y%29%285x-6y%29.


Note: you can check the answer by expanding %287x%2B9y%29%285x-6y%29 to get 35x%5E2%2B3xy-54y%5E2 or by graphing the original expression and the answer (the two graphs should be identical).