SOLUTION: A designer attempts to arrange the characters of his artwork in the form of a square grid with equal numbers of rows and columns, but finds that 24 characters are left out. When he

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Question 206593: A designer attempts to arrange the characters of his artwork in the form of a square grid with equal numbers of rows and columns, but finds that 24 characters are left out. When he tries to add one more row and column, he finds that he has 25 too few characters. Find the number of characters used by the designer.

Found 2 solutions by ankor@dixie-net.com, Marth:
Answer by ankor@dixie-net.com(22740) (Show Source):
You can put this solution on YOUR website!
A designer attempts to arrange the characters of his artwork in the form of a square grid with equal numbers of rows and columns, but finds that 24 characters are left out. When he tries to add one more row and column, he finds that he has 25 too few characters. Find the number of characters used by the designer.
:
If x = number of characters in each row
then
x*x + 24 = total number of characters
:
"When he tried to add one more row and column, he found that he was short of 25 characters."
Now, the number of rows is "x + 1", and the number of columns is "x + 1", so we have
(x + 1)(x + 1) - 25 = total number of characters.
FOIL
x^2 + 2x + 1 - 25
x^2 + 2x - 24
:
An equation from these two expression
:
x^2 + 2x - 24 = x^2 + 24
x^2 - x^2 + 2x - 24 - 24 = 0
:
x^2's cancel
2x - 48 = 0
2x = +48
x = 48/2
x = 24
:
Grid is 24 by 24 which contains 576 character + 24 left out = 600 characters
Grid of 25 by 25 = 625 character so he would have 25 blanks

Answer by Marth(57) (Show Source):
You can put this solution on YOUR website!
let x = the width of the grid.
= the number of characters
= the number of characters.
___
You can solve this algebraically, or using properties of squares, which is faster.
For any two numbers and such that ,

Here we have
, which matches the difference of two squares above.
So, x=24 and x+1 = 25.
_____
To find the number of characters, use either value in the equations obtained at the start.
= the number of characters
=
=
=600