```Question 113945
"A designer, attempting to arrange the characters of his artwork in the form of a square grid with equal number of row and columns, found that 24 characters were left out."
:
If x = number of characters in each row
we translate this statement as
x*x + 24 = total number of characters
which is
x^2 + 24
:
"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.
Which is:
(x^2 + 2x + 1 - 25)
:
The two cases equal the same number characters
:
Add a row - 25 = original grid + 24
x^2 + 2x + 1 -25 = x^2 + 24
:
x^2 + 2x -24 = x^2 + 24
:
x^2 - x^2 + 2x - 24 - 24 = 0
:
2x - 48 = 0
:
2x = +48
:
x = 48/2
:
x = 24
:
Grid is 24 by 24 which contains 576 character + 24 left out = 600
:
Grid of 25 by 25 = 625 character so he would have 25 blanks
:
:
2). Some students planned for a get-together. The budget for food was \$500. Five of the students failed to come, and therefore the cost of food for each member increased by \$5. How many students attended the get-together?
:
Let x = number of students that actually attended
Then
(x+5) = number of students that were planned to attend
:
Actual cost of food per student = 500/x
:
Planned cost for food per student = 500/(x+5)
:
Planned cost + 5\$ = Actual cost
{{{500/((x+5))}}} + 5 = {{{500/x}}}
:
Multiply equation by x(x+5) to get rid of the denominators
x(x+5)*{{{500/((x+5))}}} + x(x+5)*(5) = x(x+5)*{{{500/x}}}
:
Cancel out the denominators and you have:
500x + 5(x^2+5x) = 500(x+5)
:
500x + 5x^2 + 25x = 500x + 2500
:
Arrange like terms on the left
5x^2+ 500x - 500x + 25x - 2500 = 0
:
5x^2 + 25x - 2500 = 0
:
Simplify, divide equation by 5
x^2 + 5x - 500 = 0
:
Could this be the quadratic equation that you were seeking?

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