an algebra problem: the product of the page numbers on 2 facing pages of a book is 4160; what is the equation

Question

i can only come up with X * (X + 1) = 4160, which is pretty difficult for 7th grade algebra.

Answer 1

The equation is indeed X * (X + 1) = 4160, but you don't need so much algebra to get the answer.

1. The pages are WHOLE NUMBERS, not fractions.
2. One page is EVEN, and the other page is ODD.

4160 = 2 * 2 * 2 * 2 * 2 * 2 * 5 * 13, and the odd page must have only odd factors, so there are only 3 choices for it - page 5, page 13, or page 65. Get the even page by division, or by multiplying up the other factors. Surely that's okay for 7th grade?

Otherwise, from the difficult equation, go like this:
X^2 + X = 4160
4X^2 + 4X = 16640, by multiplying by 4
4X^2 + 4X + 1 = 16641, by adding 1 to each side
(2x + 1)^2 = 129^2, by taking exact square root of each side
2X + 1 = 129, 2X = 128, X = 64
but you decide whether that's 7th grade or not.

Answer 2

Yes it is tough, because you need the quadratic equation solution, as x(x +1) = 4160, forms to quadratic equation x^2 + x - 4160 = 0 will give you a solution of 64 for x. You'll get lots of help using that approach so I will suggest something else:

Want to see a trick? Take the square root of 4160 to get started, this is 64.5 ! (IF you can use a calculator) You know the two numbers have to be consecutive integers beginning with an even number (as in an open 2-pages of a book), so try 64 and 65, the numbers around the 64.5 .... 64 x 65 = 4160.

Answer 3

x(x + 1) = 4160

x^2 + x = 4160
x^2 + x - 4160 = 0

x = (-b ± sqrt(b^2 - 4ac))/(2a)

x = (-1 ± sqrt(1^2 - 4(1)(-4160)))/(2(1))
x = (-1 ± sqrt(1 + 16640))/2
x = (-1 ± sqrt(16641))/2
x = (-1 ± 129)/2
x = (-130/2) or (128/2)
x = -65 or 64

since you can't have negative pages

ANS :

The pages are 64 and 65

Answer 4

x*(x+1)=4160
x^2+x-4160 = 0
(x-64)(x+65)=0
x=64 (since it can't be -65)

There is a simpler way... x*(x+1) ~ x^2
so find the square root of 4160 which is about 64.5
thus you know x=64

Answer 5

The question is totally false, as two facing pages will always have an uneven (odd numbered) sum

Answer 6

x sqared equals 4159

Answer 7

sorry but i can't help you there