While finding the amount of seed needed to plant his three square wheat fields, Hank observed that the side of one field was 1 kilometer longer than the side of the smallest field and that the side of the largest field was 3 kilometers longer than the smallest field. If the total area of the three fields is 38 square kilometers, than what is the area of each field?

x^2 + (x+1)^2 + (x+3)^2 = 38

expand, simplify, then solve the quadratic equation.

reject one of the answers which will be negative.

s = side of smallest field

(s+1) = side of middle fiels
(s+3) = side of biggest fiels
so
s^2 + (s+1)^2 + (s+3)^2 - 38 = 0

s^2 + s^2 + 2 s + 1 + s^2 +6 s + 9 - 38 = 0

3 s^2 + 8 s - 28 = 0

(3s+14)(s-2) = 0
s = 2 is positive answer
area of small field = 4
then
s+1 = 3
area of middle size field = 9
s+3 = 5
area of big field = 25

To solve this problem, we can use algebraic equations and a system of equations.

Let's say the side length of the smallest field is "x" kilometers.

According to the problem, the side length of the second field is 1 kilometer longer than the smallest field, so its side length is "x + 1" kilometers.

Similarly, the side length of the largest field is 3 kilometers longer than the smallest field, so its side length is "x + 3" kilometers.

The area of a square is calculated by squaring its side length. Therefore, the area of each field is given by the formula: Area = side length^2.

For the smallest field: Area = x^2
For the second field: Area = (x + 1)^2
For the largest field: Area = (x + 3)^2

The sum of the areas of all three fields is given as 38 square kilometers:
x^2 + (x + 1)^2 + (x + 3)^2 = 38

Now, we can solve this equation to find the value of "x" and then calculate the area of each field.

Expanding and simplifying the equation:
x^2 + x^2 + 2x + 1 + x^2 + 6x + 9 = 38
3x^2 + 8x + 10 = 38
3x^2 + 8x - 28 = 0

Next, we can solve this quadratic equation either by factoring, completing the square, or using the quadratic formula.

By factoring, we can simplify the equation:
(3x - 4)(x + 7) = 0

Setting each factor equal to zero:
3x - 4 = 0 or x + 7 = 0

Solving for "x":
3x = 4 or x = -7
x = 4/3 or x = -7

Since we cannot have a negative side length for a field, we discard the solution x = -7.

Therefore, the smallest field has a side length of x = 4/3 kilometers.

To find the areas of the fields, we substitute this value back into our formulas:

Area of the smallest field = (4/3)^2 = 16/9 square kilometers

Area of the second field = (4/3 + 1)^2 = (7/3)^2 = 49/9 square kilometers

Area of the largest field = (4/3 + 3)^2 = (13/3)^2 = 169/9 square kilometers

So, the area of each field is:
Smallest field: 16/9 square kilometers
Second field: 49/9 square kilometers
Largest field: 169/9 square kilometers