A student opened her math book and computed the sum of the numbers on two facing pages. Then she turned to the next page and computed the sum of the numbers on these two facing pages. Finally, she computed the product of the two sums, and her calculator displayed the number 80,085. What were the four page numbers?

The four page numbers were 143, 144, 145 and 146.

Step-by-step explanation:

We can assume the first page as x and the next page as x + 1.

The next two facing pages would similarly be x+2 and x+3.

The sum of the numbers on the first two facing pages can be calculated as:

Sum of first two pages = x + (x + 1)

Sum of first two pages = 2x + 1

Similarly, the sum of the next two page numbers is:

Sum of next two pages = (x+2) + (x+3)

Sum of next two pages = 2x + 5

Then, we need to compute the product of the two sums.

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

= (2x) (2x) + (2x) (5) + (1) (2x) + (1) (5)

= 4x² + 10x + 2x + 5

= 4x² + 12x + 5

This product is equal to 80085 so,

4x² + 12x + 5 = 80085

4x² + 12x + 5 - 80085 = 0

4x² + 12x - 80080 = 0

we can solve this equation using the quadratic formula to find out the value of x. So here we have:

a = 4, b = 12, c = - 80080

Using the quadratic formula:

(-b±√b²-4ac) / 2a

= (-12±√ (12) ²-4 (4) (-80080)) / 2 (4)

= (-12±√144 - (-128120)) / 8

= (-12±√1281424) / 8

= (-12±1132) / 8

= (12 + 1132) / 8 or (12-1132) / 8

x = 143 or - 140

Since the value of x is a page number hence it can not be negative. So, we will choose the positive value i. e. 143.

We initially supposed x as the first page which the student opened. So, the first page number is 143. The next pages are:

x+1 = 143 + 1 = 144

x+2 = 143 + 2 = 145

x+3 = 143 + 3 - 146

So, the four page numbers were 143, 144, 145 and 146.