SOLUTION: A square vegetable garden is to be tilled and then enclosed with a fence. If the fence costs $4.00 per foot and the cost of preparing the soil is $1.00 per ft2, determine the size

Question 903553: A square vegetable garden is to be tilled and then enclosed with a fence. If the fence costs $4.00 per foot and the cost of preparing the soil is $1.00 per ft2, determine the size of the garden that can be enclosed for $561.00
Answer by richwmiller(17219) (Show Source): You can put this solution on YOUR website!
perimeter is 4s area is s^2
4s*4+1*s^2=561
16s+s^2=561
rearrange
s^2+16s-561=0

Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

Looking at the expression , we can see that the first coefficient is , the second coefficient is , and the last term is .

Now multiply the first coefficient by the last term 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 :

1,3,11,17,33,51,187,561

-1,-3,-11,-17,-33,-51,-187,-561

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

These factors pair up and multiply to .

1*(-561) = -561
3*(-187) = -561
11*(-51) = -561
17*(-33) = -561
(-1)*(561) = -561
(-3)*(187) = -561
(-11)*(51) = -561
(-17)*(33) = -561

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	-561	1+(-561)=-560
3	-187	3+(-187)=-184
11	-51	11+(-51)=-40
17	-33	17+(-33)=-16
-1	561	-1+561=560
-3	187	-3+187=184
-11	51	-11+51=40
-17	33	-17+33=16

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. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.

Combine like terms. Or factor out the common term

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

Answer:

So factors to .

In other words, .

Note: you can check the answer by expanding to get or by graphing the original expression and the answer (the two graphs should be identical).

Solved by pluggable solver: COMPLETING THE SQUARE solver for quadratics

Read this lesson on completing the square by prince_abubu, if you do not know how to complete the square.
Let's convert to standard form by dividing both sides by 1:
We have: . What we want to do now is to change this equation to a complete square . How can we find out values of somenumber and othernumber that would make it work?
Look at : . Since the coefficient in our equation that goes in front of s is 16, we know that 16=2*somenumber, or . So, we know that our equation can be rewritten as , and we do not yet know the other number.
We are almost there. Finding the other number is simply a matter of not making too many mistakes. We need to find 'other number' such that is equivalent to our original equation .

The highlighted red part must be equal to -561 (highlighted green part).

, or .
So, the equation converts to , or .

Our equation converted to a square , equated to a number (625).

Since the right part 625 is greater than zero, there are two solutions:

, or

Answer: s=17, -33.

discard negative answers
s=17

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