I've been staring at this problem for about 15 minutes, and it's driving me nuts, because I know the answer, and I know the formula, but I don't know how to get the answer out of the formula:
The Orchard brothers always plant their apple tress in square arrays. This year they planted 31 more apple trees in their square orchard than last year. If the orchard is still square, how many apple trees are in the orchard this year? (It shows a picture of square arrays of 1, 4, and 9.) Continuing with this logic, I figured that each year adds another perfect square, so 16, 25, 36, all the way up to 256, which is 31 more than 225. I figured that out just by writing out all the squares up to that point. Now, if n is the number of years, and n squared is the number of trees, then I thought the formula would be n squared = (n-1)squared + 31. If you plug 16 in as n, this works. However, I can't seem to SOLVE for n effectively. I keep getting n squared = squared + 1 + 31, and if I subtract n squared from both sides, I get 0 = 32. Can I not subtract the n squared? I tried taking the square root of n squared, but that just leaves me with n = n + (squareroot)32, which becomes 0 = (squareroot)32. It's not a big deal since I solved it anyway, but I'd just like to know how to solve it algebraically in case it comes up on my final with bigger numbers where I can't just write out the terms. Thanks so much!
The Orchard brothers always plant their apple tress in square arrays. This year they planted 31 more apple trees in their square orchard than last year. If the orchard is still square, how many apple trees are in the orchard this year? (It shows a picture of square arrays of 1, 4, and 9.) Continuing with this logic, I figured that each year adds another perfect square, so 16, 25, 36, all the way up to 256, which is 31 more than 225. I figured that out just by writing out all the squares up to that point. Now, if n is the number of years, and n squared is the number of trees, then I thought the formula would be n squared = (n-1)squared + 31. If you plug 16 in as n, this works. However, I can't seem to SOLVE for n effectively. I keep getting n squared = squared + 1 + 31, and if I subtract n squared from both sides, I get 0 = 32. Can I not subtract the n squared? I tried taking the square root of n squared, but that just leaves me with n = n + (squareroot)32, which becomes 0 = (squareroot)32. It's not a big deal since I solved it anyway, but I'd just like to know how to solve it algebraically in case it comes up on my final with bigger numbers where I can't just write out the terms. Thanks so much!
