SOLUTION: The Lassalle’s have an orange grove that contains 90 trees. The number of trees in each row is 3 more than twice the number of rows. Find the number of rows and the number of tree

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Question 885700: The Lassalle’s have an orange grove that contains 90 trees. The number of trees in each row is 3 more than twice the number of rows. Find the number of rows and the number of trees per row.
Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
90=(2x+3)*x
2x^2+3x-90=0
we want factors of -180 which add to 3
x=6
2x+3=15
6 rows with 15 trees in each row.
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression 2x%5E2%2B3x-90, we can see that the first coefficient is 2, the second coefficient is 3, and the last term is -90.



Now multiply the first coefficient 2 by the last term -90 to get %282%29%28-90%29=-180.



Now the question is: what two whole numbers multiply to -180 (the previous product) and add to the second coefficient 3?



To find these two numbers, we need to list all of the factors of -180 (the previous product).



Factors of -180:

1,2,3,4,5,6,9,10,12,15,18,20,30,36,45,60,90,180

-1,-2,-3,-4,-5,-6,-9,-10,-12,-15,-18,-20,-30,-36,-45,-60,-90,-180



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



These factors pair up and multiply to -180.

1*(-180) = -180
2*(-90) = -180
3*(-60) = -180
4*(-45) = -180
5*(-36) = -180
6*(-30) = -180
9*(-20) = -180
10*(-18) = -180
12*(-15) = -180
(-1)*(180) = -180
(-2)*(90) = -180
(-3)*(60) = -180
(-4)*(45) = -180
(-5)*(36) = -180
(-6)*(30) = -180
(-9)*(20) = -180
(-10)*(18) = -180
(-12)*(15) = -180


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



First NumberSecond NumberSum
1-1801+(-180)=-179
2-902+(-90)=-88
3-603+(-60)=-57
4-454+(-45)=-41
5-365+(-36)=-31
6-306+(-30)=-24
9-209+(-20)=-11
10-1810+(-18)=-8
12-1512+(-15)=-3
-1180-1+180=179
-290-2+90=88
-360-3+60=57
-445-4+45=41
-536-5+36=31
-630-6+30=24
-920-9+20=11
-1018-10+18=8
-1215-12+15=3




From the table, we can see that the two numbers -12 and 15 add to 3 (the middle coefficient).



So the two numbers -12 and 15 both multiply to -180 and add to 3



Now replace the middle term 3x with -12x%2B15x. Remember, -12 and 15 add to 3. So this shows us that -12x%2B15x=3x.



2x%5E2%2Bhighlight%28-12x%2B15x%29-90 Replace the second term 3x with -12x%2B15x.



%282x%5E2-12x%29%2B%2815x-90%29 Group the terms into two pairs.



2x%28x-6%29%2B%2815x-90%29 Factor out the GCF 2x from the first group.



2x%28x-6%29%2B15%28x-6%29 Factor out 15 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.



%282x%2B15%29%28x-6%29 Combine like terms. Or factor out the common term x-6



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Answer:



So 2%2Ax%5E2%2B3%2Ax-90 factors to %282x%2B15%29%28x-6%29.



In other words, 2%2Ax%5E2%2B3%2Ax-90=%282x%2B15%29%28x-6%29.



Note: you can check the answer by expanding %282x%2B15%29%28x-6%29 to get 2%2Ax%5E2%2B3%2Ax-90 or by graphing the original expression and the answer (the two graphs should be identical).