determine three consecutive even integers, if the square of the largest integer less the square of the middle integer is 20 less than the square of the smallest integer

so, if the numbers are x-2, x and x+2, we have

(x+2)^2 - x^2 = (x-2)^2 - 20
solve for x and you get -2,10
check:
12^2 - 10^2 = 8^2 - 20 -- yep
0^2 - 2^2 = (-4)^2 - 20 -- yep

Let's break down the problem step by step:

Step 1: Let's assume the smallest even integer is x.

Step 2: The next consecutive even integer would be (x + 2).

Step 3: The largest even integer would be (x + 4).

Step 4: According to the problem statement, the square of the largest integer minus the square of the middle integer is 20 less than the square of the smallest integer. Mathematically, this can be expressed as ((x + 4)^2 - (x + 2)^2 = x^2 - 20).

Step 5: Simplifying the equation:
(x^2 + 8x + 16 - x^2 - 4x - 4) = x^2 - 20
(4x + 12) = x^2 - 20

Step 6: Rearranging the equation:
x^2 - 4x - 32 = 0

Step 7: Factoring the equation:
(x - 8)(x + 4) = 0

Step 8: Solving for x:
x - 8 = 0 -> x = 8 or x + 4 = 0 -> x = -4

Since the problem states that we need to find even integers, we only consider the positive value of x.

Therefore, the three consecutive even integers are:
- The smallest even integer is x = 8.
- The middle even integer is (x + 2) = 10.
- The largest even integer is (x + 4) = 12.

To solve this problem, let's assume the three consecutive even integers are x, x + 2, and x + 4.

According to the given conditions, we can set up the following equation:

(x + 4)^2 - (x + 2)^2 = (x)^2 - 20

Now, let's simplify this equation step by step:

Expanding the squares on both sides:

(x^2 + 8x + 16) - (x^2 + 4x + 4) = x^2 - 20

Simplifying further:

x^2 + 8x + 16 - x^2 - 4x - 4 = x^2 - 20

Combine like terms:

4x + 12 = x^2 - 20

Rearrange the equation:

x^2 - 4x - 32 = 0

Now we have a quadratic equation. Let's solve it by factoring:

(x - 8)(x + 4) = 0

By setting each factor equal to zero, we get two possible values for x:

x - 8 = 0 or x + 4 = 0

Solving these equations:

x = 8 or x = -4

Since we are looking for even integers, we can disregard the negative value. Therefore, x = 8.

So the consecutive even integers are 8, 10, and 12.

To verify our solution, let's substitute these values back into the original equation:

(12^2) - (10^2) = (8^2) - 20

144 - 100 = 64 - 20

44 = 44

Both sides of the equation are equal, confirming that our solution is correct.