help with quadratic equations?

Question

A designer attempting to arrange the characters of his artwork in the form of a square grid with equal numbers in a row and columns found that 24 characters were left out. When he tried to add one more row and column he found he was short 25 characters. Can you find the number of characters used by the designer?
This is complicated to me please explain how to get a quadratic equation from this and what the number of characters are used by the designer

Answer 1

Let x be the number of characters & y be the number of characters in each row/column in first attempt.

y^2 + 24 = x.........(1)

(y+1)^2 - 25 = x..........(2)

from (1) & (2)
y^2 + 24 = (y+1)^2 - 25
y^2 + 24 = y^2 + 1 +2y -25
24=2y -24 ....(y^2 get cancelled)
2y=48
y=48/2=24

Put it in (1)
(24)^2 + 24 = x
600=x

Number of characters = 600.
In first attempt he tried 24 x 24 & he left with 24. In second attempt he tried 25 x 25 & he fell 25 short.

Answer 2

The characters are put into a square grid. Since we don't know what the size is we shall call it NxN (N rows and N columns of characters). The total number of characters in the grid is then N times N or N squared (N**2).
We also don't know how many characters are in the artwork so let us say there are M characters in it.
When put into this square grid there are 24 characters left out so: M = N**2 + 24 (the total number of characters is equal to the number of characters in the grid plus the 24 that were left out).
Add a row and column (so N becomes N+1) and you now have a grid with 25 spaces that are not filled in. So, like the previous one: M = (N+1)**2 - 25.
This becomes: M = N**2 +2*N + 1 - 25 = N**2 + 2*N + 24
Now subtract these two expressions for M.
[N**2 + 2*N + 24] - [N**2 + 24] = M - M = 0
2*N - 48 = 0 and 2*N = 48 giving N = 24
So the total number of characters is M = (24)**2 + 24 or 600.

Answer 3

well he has 600 characters. if you have a grid that is 24x24 you will use 576 characters, while a grid that is 25x25 will use 625 characters. if he has 600 characters, 576 will leave you with 24 left over, and 625 will be 25 short. the problem is, im not sure how you would make a quadratic equation from this because a quadratic equation will have 2 real solutions. the only solution here is 600. but hopefully that will get you on the right track.