Okay, let's tackle this problem step by step. The question says that the product of two consecutive positive even integers is 288, and we need to find the greater of the two integers. Hmm, consecutive even integers... so they are two even numbers that follow each other. Like 2 and 4, 4 and 6, etc. Since they are consecutive even numbers, the difference between them is 2. 

First, let me denote the smaller integer as x. Then the next consecutive even integer would be x + 2. The product of these two numbers is given as 288. So, mathematically, that would be x * (x + 2) = 288. Let me write that equation down:

x(x + 2) = 288

Expanding the left side, we get:

x² + 2x = 288

To solve for x, I can rearrange this into a standard quadratic equation form. Subtract 288 from both sides:

x² + 2x - 288 = 0

Now, I need to solve this quadratic equation. Quadratic equations can be solved using factoring, completing the square, or the quadratic formula. Let me see if this equation factors nicely. The quadratic is x² + 2x - 288. I need two numbers that multiply to -288 and add up to 2. Let's think about the factors of 288. 

Factors of 288: 1 & 288, 2 & 144, 3 & 96, 4 & 72, 6 & 48, 8 & 36, 9 & 32, 12 & 24, 16 & 18. Since the product is negative, one number has to be positive and the other negative. The sum is 2, which is positive. So the positive number should be larger. Let's check the pairs:

Looking for two numbers where one is larger than the other by 2 (since the middle term is 2x). Let's check 16 and 18: 18 - 16 = 2. But 16 * 18 = 288. But in our equation, the product is -288, so maybe one is positive and the other negative. So if we take 18 and -16, their product is -288 and their sum is 2. Let's check: 18 + (-16) = 2. Yes, that works. So the factors would be (x + 18)(x - 16) = 0. Therefore, the solutions are x = -18 or x = 16. But since the problem states that the integers are positive, x must be 16. So the two consecutive even integers are 16 and 18. The greater one is 18. 

Wait, let me verify that. If x is 16, then x + 2 is 18. 16*18 is indeed 288. Correct. So the answer should be 18. 

But just to make sure I didn't make a mistake in factoring, let me check using the quadratic formula. The quadratic equation is x² + 2x - 288 = 0. The quadratic formula is x = [-b ± sqrt(b² - 4ac)] / (2a). Here, a = 1, b = 2, c = -288. Plugging in:

Discriminant: b² - 4ac = 4 - 4*1*(-288) = 4 + 1152 = 1156

Square root of 1156 is 34. So,

x = [-2 ± 34]/2

Calculating both solutions:

x = (-2 + 34)/2 = 32/2 = 16

x = (-2 - 34)/2 = -36/2 = -18

Same results. Since we need positive integers, x = 16, so the next even integer is 18. Therefore, the greater integer is 18. 

Alternatively, maybe there's another way to approach this problem without algebra. Let's think. If the product of two consecutive even numbers is 288, then taking the square root of 288 might give us an approximate midpoint. The square root of 288 is approximately 16.97. Since they are consecutive even numbers, they should be around 16 and 18. Indeed, 16*18=288. So that's another way to see it. 

But wait, why would the square root be near the midpoint? Because if two numbers multiply to a certain product, their geometric mean is the square root. So the numbers should be around that value. Since they are consecutive even numbers, the midpoint would be between them. For example, if the numbers were 16 and 18, their average is 17, which is close to the square root of 288 (16.97). So that makes sense. 

Alternatively, maybe we can use trial and error with even numbers. Let's try 14*16=224, which is too low. 16*18=288. There we go. So that's the pair. 

Therefore, the greater integer is 18. I think that's solid. Let me just check once more. 16*18: 10*18=180, 6*18=108, 180+108=288. Correct. Yep, that's right.
Thus, the greater of the two consecutive positive even integers whose product is 288 is $\boxed{18}$.
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