Nick was surveying his field and saw some cows and ducks. Nick counted a total of 80 heads and 248 legs. Assuming there are no mutations in the animals, determine exactly how many ducks and
cows he saw. Let X represent the number of ducks and y represent the cows.
Total Equation:
Legs Equation:
How many ducks?
How many cows?

44 cows and 36 ducks.

Step-by-step explanation:

x is ducks and y is cows.

Assuming both have 1 head each, we can write:

x + y = 80.

Let's assume each x(cow) has 4 legs and each y(duck) has two legs.

Then we can add the numbers of legs:

2x + 4y = 248 legs

Now rearrange x + y = 80 to:

x = 80-y

and substitute:

2x + 4y = 248 legs

2(80-y) + 4y = 248 legs

160 - 2y + 4y = 248 legs

2y = 88 legs

y = 44   [There are 44 cows]

Use this value to find the number of ducks:

x + y = 80

44 + y = 80

y = 36  [There are 36 ducks]

Check:

Do 44 cows and 36 ducks have a total of 248 legs?

4*44 + 2*36 = 248 ?

176 + 72 = 248?

YES

tell you what. i'll find the length of the medians and you can fill in the blanks where they belong.

br

b = (2a,0)

r = (a/2,b/2) remember these things are medians. they go to the 1/2 way point of the line opposite the vertex they face.

br^2 =   (2a - a/2)^2 + (0 - b/2)^2

br^2 = (3/2 a) ^2 + b^2 / 4

br^2 = 9/4 a^2 + b^2 / 4   we need to find some relationship between a and b.

let's try ab = bc  

ab = 2a

bc = (2a - a)^2 + (b - 0)^2

bc = sqrt(a^2 + b^2)

ab = bc

2a = sqrt(a^2 + b^2) square both sides.

4a^2 = a^2 + b^2 subtract a^2 from both sides.

sqrt(3a^2) = sqrt(b^2)

sqrt(3)a = b

let's put b into br^2

br^2 = 9/4 a^2 + 3b^2 / 4

br^2 = 12 a^2 / 4

br^2 = 3a^2 

br = sqrt(3) * a

cp

c = (a,b) ; p = (a,0) this is another application of the distance formula, and it is a good one.

cp^2 = (a - a)^2 + (b - 0)^2

cp^2 = 0 + b^2

cp^2 = b^2 which you can see from the diagram. 

cp = sqrt(b^2)

cp = b but b = sqrt(3) * a 

cp = sqrt(3)*a

aq

a = (0,0)

q = (3/2) a, b/2

aq^2 = (3a/2 -0)^2 + (b/2 - 0)^2

aq^2 = 9a^2/4 + b^2/4

but b = sqrt(3) * a 

aq^2 = 9a^2 / 4 + (sqrt(3)*a)^2 /4

aq^2 = 9 a^2/ 4 + 3 a^2 / 4

aq^2 = 12 a^2/4

aq^2 = 3 a^2

aq = (sqrt  3) * a which agrees with the other two.

answer: (x+2) (x+3) are the binomials for your question : )