SOLUTION: factor 22nsquare+n-5 also factor 14zsquare-49z+35 And factor 5xsquare+17x+6

Algebra ->  Equations -> SOLUTION: factor 22nsquare+n-5 also factor 14zsquare-49z+35 And factor 5xsquare+17x+6      Log On


   

Question 551194: factor 22nsquare+n-5
also factor 14zsquare-49z+35

And factor 5xsquare+17x+6
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'll do the first one to get you started



Looking at the expression 22n%5E2%2Bn-5, we can see that the first coefficient is 22, the second coefficient is 1, and the last term is -5.


Now multiply the first coefficient 22 by the last term -5 to get %2822%29%28-5%29=-110.


Now the question is: what two whole numbers multiply to -110 (the previous product) and add to the second coefficient 1?


To find these two numbers, we need to list all of the factors of -110 (the previous product).


Factors of -110:
1,2,5,10,11,22,55,110
-1,-2,-5,-10,-11,-22,-55,-110


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


These factors pair up and multiply to -110.
1*(-110) = -110
2*(-55) = -110
5*(-22) = -110
10*(-11) = -110
(-1)*(110) = -110
(-2)*(55) = -110
(-5)*(22) = -110
(-10)*(11) = -110

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 1:


First NumberSecond NumberSum
1-1101+(-110)=-109
2-552+(-55)=-53
5-225+(-22)=-17
10-1110+(-11)=-1
-1110-1+110=109
-255-2+55=53
-522-5+22=17
-1011-10+11=1



From the table, we can see that the two numbers -10 and 11 add to 1 (the middle coefficient).


So the two numbers -10 and 11 both multiply to -110 and add to 1


Now replace the middle term 1n with -10n%2B11n. Remember, -10 and 11 add to 1. So this shows us that -10n%2B11n=1n.


22n%5E2%2Bhighlight%28-10n%2B11n%29-5 Replace the second term 1n with -10n%2B11n.


%2822n%5E2-10n%29%2B%2811n-5%29 Group the terms into two pairs.


2n%2811n-5%29%2B%2811n-5%29 Factor out the GCF 2n from the first group.


2n%2811n-5%29%2B1%2811n-5%29 Factor out 1 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.


%282n%2B1%29%2811n-5%29 Combine like terms. Or factor out the common term 11n-5


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


Answer:


So 22n%5E2%2Bn-5 factors to %282n%2B1%29%2811n-5%29.


In other words, 22n%5E2%2Bn-5=%282n%2B1%29%2811n-5%29.


Note: you can check the answer by expanding %282n%2B1%29%2811n-5%29 to get 22n%5E2%2Bn-5 or by graphing the original expression and the answer (the two graphs should be identical).