SOLUTION: I am having trouble figuring out how to turn these wrod problems into quadratic equations. Can anyone help? 1). a designemr attempting to arranfe the characters of his artwork in

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Question 113945: I am having trouble figuring out how to turn these wrod problems into quadratic equations. Can anyone help?
1). a designemr attempting to arranfe the characters of his artwork in the form of a square grid with equal number of rows and columns, found that 24 characters were left out. When he tried to add one more row and column, he found that he was short 25 characters. Can you find the number of characters used by the designer? I know that he has 600 characters but, I'm not sure how to transform it into a quadratic equation.
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? I know that 20 students attended, but don't know how to set up the quadratic equation.
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
"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%2F%28%28x%2B5%29%29 + 5 = 500%2Fx
:
Multiply equation by x(x+5) to get rid of the denominators
x(x+5)*500%2F%28%28x%2B5%29%29 + x(x+5)*(5) = x(x+5)*500%2Fx
:
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?