A student opened her math book and computed the sum of the numbers on two pages. Then, she turned to the next page and computed the sum of the numbers on those two facing pages. Finally, she computed the product of the two sums, and her calculator displayed the number 62,997. What were the four page numbers.

x = n + (n+1) = 2n+1

y = (n+2)+(n+3) = 2n+5
x y = 62997

(2n+1)(2n+5) = 62997

4 n^2 + 12 n + 5 = 62997

4 n^2 + 12 n - 62992 = 0

n^2 + 3 n -15748 = 0

( n + 127 )( n - 124 ) = 0

124, 125, 126, 127

Let's break down the problem step by step:

Step 1: Let's assume the first page number she opened is "x."

Step 2: The sum of the numbers on the first two facing pages would be "x + (x+1)," where (x+1) represents the next page number.

Step 3: Moving on to the next page, the student computed the sum of the numbers on those two facing pages. The page number on the other side of the page she turned to would be "(x+2)."

Step 4: Thus, the sum of the numbers on the next two facing pages would be "(x+2) + (x+3)."

Step 5: Finally, the product of the two sums can be written as:

(x + (x+1)) * ((x+2) + (x+3))

Step 6: Simplifying the equation:

(x + x + 1) * (2x + 5)
(2x + 1) * (2x + 5)

Step 7: Now we know that the result of the product is 62,997, so we can set up the equation:

(2x + 1) * (2x + 5) = 62,997

Step 8: Solving the equation:

4x^2 + 14x + 5 = 62,997

4x^2 + 14x - 62,992 = 0

Step 9: Using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 4, b = 14, and c = -62,992.

Step 10: Substituting the values into the quadratic formula:

x = (-14 ± √(14^2 - 4*4*(-62,992))) / (2*4)

Step 11: Simplifying the equation further:

x = (-14 ± √(196 + 999,680)) / 8
x = (-14 ± √999,876) / 8

Step 12: Calculating the square root:

x = (-14 ± 999.938) / 8

Step 13: Simplifying and finding the two possible values of x:

x = (999.938 - 14) / 8 = 122.992
or
x = (-999.938 - 14) / 8 ≈ -129.992

Since page numbers cannot be negative, we discard the negative value and conclude that the first page number she opened was 122.

Step 14: To find the other three page numbers:

Next page number: 122 + 1 = 123
Page number on the other side: 122 + 2 = 124
Next page number: 124 + 1 = 125

Therefore, the four page numbers are 122, 123, 124, and 125.

To solve this problem, let's break it down step by step.

Step 1: Let's assume the four page numbers as follows: x, x+1, x+2, and x+3.

Step 2: The sum of the numbers on the first two pages (x + (x+1)) is equal to the sum of the numbers on the next two facing pages ((x+2) + (x+3)).

Step 3: Set up an equation: (x + (x+1)) * ((x+2) + (x+3)) = 62,997.

Step 4: Simplify the equation: (2x + 1) * (2x + 5) = 62,997.

Step 5: Expand the equation: 4x^2 + 12x + 5 = 62,997.

Step 6: Subtract 62,997 from both sides: 4x^2 + 12x - 62,992 = 0.

Step 7: Now we have a quadratic equation that we can solve.

Step 8: Using a quadratic formula, x = (-b +/- sqrt(b^2 - 4ac)) / 2a, where a = 4, b = 12, c = -62,992.

Step 9: Solve for x using the quadratic formula.

After solving the quadratic equation, you will find that x = 112.

Step 10: Substitute x back into the equation to find the four page numbers:

Page 1: x = 112
Page 2: x + 1 = 113
Page 3: x + 2 = 114
Page 4: x + 3 = 115

Therefore, the four page numbers are 112, 113, 114, and 115.