Question 144532
He has n characters.
If he makes it x by x, he has 24 left over, so n = x^2+24
If he makes it (x+1) by (x+1), he's 25 short, so n = (x+1)^2 - 25
{{{n = (x+1)^2 - 25}}}
{{{n = x^2+24}}}
{{{x^2+24 = (x+1)^2 - 25}}}  Since both equal n
{{{x^2+24 = x^2+2x+1 - 25}}}
{{{24 = 2x-24}}}
2x = 48
x = 24

n = 24*24 + 24 = 600
25*25 is 625, so he would be 25 short.

There was no need to solve a quadratic.