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

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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) About Me  (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 s%5E2%2B16s-561, we can see that the first coefficient is 1, the second coefficient is 16, and the last term is -561.



Now multiply the first coefficient 1 by the last term -561 to get %281%29%28-561%29=-561.



Now the question is: what two whole numbers multiply to -561 (the previous product) and add to the second coefficient 16?



To find these two numbers, we need to list all of the factors of -561 (the previous product).



Factors of -561:

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 -561.

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 16:



First NumberSecond NumberSum
1-5611+(-561)=-560
3-1873+(-187)=-184
11-5111+(-51)=-40
17-3317+(-33)=-16
-1561-1+561=560
-3187-3+187=184
-1151-11+51=40
-1733-17+33=16




From the table, we can see that the two numbers -17 and 33 add to 16 (the middle coefficient).



So the two numbers -17 and 33 both multiply to -561 and add to 16



Now replace the middle term 16s with -17s%2B33s. Remember, -17 and 33 add to 16. So this shows us that -17s%2B33s=16s.



s%5E2%2Bhighlight%28-17s%2B33s%29-561 Replace the second term 16s with -17s%2B33s.



%28s%5E2-17s%29%2B%2833s-561%29 Group the terms into two pairs.



s%28s-17%29%2B%2833s-561%29 Factor out the GCF s from the first group.



s%28s-17%29%2B33%28s-17%29 Factor out 33 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.



%28s%2B33%29%28s-17%29 Combine like terms. Or factor out the common term s-17



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



Answer:



So s%5E2%2B16%2As-561 factors to %28s%2B33%29%28s-17%29.



In other words, s%5E2%2B16%2As-561=%28s%2B33%29%28s-17%29.



Note: you can check the answer by expanding %28s%2B33%29%28s-17%29 to get s%5E2%2B16%2As-561 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 1s%5E2%2B16s%2B-561=0 to standard form by dividing both sides by 1:
We have: 1s%5E2%2B16s%2B-561=0. What we want to do now is to change this equation to a complete square %28s%2Bsomenumber%29%5E2+%2B+othernumber. How can we find out values of somenumber and othernumber that would make it work?
Look at %28s%2Bsomenumber%29%5E2: %28s%2Bsomenumber%29%5E2+=+s%5E2%2B2%2Asomenumber%2Ax+%2B+somenumber%5E2. Since the coefficient in our equation 1s%5E2%2Bhighlight_red%28+16%29+%2A+s%2B-561=0 that goes in front of s is 16, we know that 16=2*somenumber, or somenumber+=+16%2F2. So, we know that our equation can be rewritten as %28s%2B16%2F2%29%5E2+%2B+othernumber, 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 %28s%2B16%2F2%29%5E2+%2B+othernumber is equivalent to our original equation 1s%5E2%2B16s%2Bhighlight_green%28+-561+%29=0.


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

16%5E2%2F4+%2B+othernumber+=+-561, or othernumber+=+-561-16%5E2%2F4+=+-625.
So, the equation converts to %28s%2B16%2F2%29%5E2+%2B+-625+=+0, or %28s%2B16%2F2%29%5E2+=+625.

Our equation converted to a square %28s%2B16%2F2%29%5E2, equated to a number (625).

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

system%28+%28s%2B16%2F2%29+=+%2Bsqrt%28+625+%29%2C+%28s%2B16%2F2%29+=+-sqrt%28+625+%29+%29
, or

system%28+%28s%2B16%2F2%29+=+25%2C+%28s%2B16%2F2%29+=+-25+%29
system%28+s%2B16%2F2+=+25%2C+s%2B16%2F2+=+-25+%29
system%28+s+=+25-16%2F2%2C+s+=+-25-16%2F2+%29

system%28+s+=+17%2C+s+=+-33+%29
Answer: s=17, -33.

discard negative answers
s=17