document.write( "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. \n" ); document.write( "

Algebra.Com's Answer #535409 by richwmiller(17219) $\"\"$ $\"About$

You can put this solution on YOUR website!
90=(2x+3)*x
\n" ); document.write( "2x^2+3x-90=0
\n" ); document.write( "we want factors of -180 which add to 3
\n" ); document.write( "x=6
\n" ); document.write( "2x+3=15
\n" ); document.write( "6 rows with 15 trees in each row.\r
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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 $\"3\"$ :

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First Number	Second Number	Sum
1	-180	1+(-180)=-179
2	-90	2+(-90)=-88
3	-60	3+(-60)=-57
4	-45	4+(-45)=-41
5	-36	5+(-36)=-31
6	-30	6+(-30)=-24
9	-20	9+(-20)=-11
10	-18	10+(-18)=-8
12	-15	12+(-15)=-3
-1	180	-1+180=179
-2	90	-2+90=88
-3	60	-3+60=57
-4	45	-4+45=41
-5	36	-5+36=31
-6	30	-6+30=24
-9	20	-9+20=11
-10	18	-10+18=8
-12	15	-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).

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