SOLUTION: The members of the theatre club were assessed equally to raise $400 for the purchase of new costumes. When four more people joined the club, the per member assessment was reduced b

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Question 950250: The members of the theatre club were assessed equally to raise $400 for the purchase of new costumes. When four more people joined the club, the per member assessment was reduced by $5.00. What was the size of the club?
Answer by macston(5194) About Me  (Show Source):
You can put this solution on YOUR website!
x=original members; a=original assessment;
ax=$400
x=$400/a
(x+4)(a-$5)=$400 Set equations equal.
(x+4)(a-$5)=ax Multiply.
ax+4a-$5x-$20=ax Subtract ax from each side.
4a-$5x=$20 Substitute for x
4a-$5($400/a)=$20
4a-$2000/a=$20 Subtract 4a from each side
-$2000/a=$20-4a Multiply each side by a
-$2000=$20a-4a^2 Add 2000 to each side
0=-4a^2+20a+2000
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation aa%5E2%2Bba%2Bc=0 (in our case -4a%5E2%2B20a%2B2000+=+0) has the following solutons:

a%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%2820%29%5E2-4%2A-4%2A2000=32400.

Discriminant d=32400 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-20%2B-sqrt%28+32400+%29%29%2F2%5Ca.

a%5B1%5D+=+%28-%2820%29%2Bsqrt%28+32400+%29%29%2F2%5C-4+=+-20
a%5B2%5D+=+%28-%2820%29-sqrt%28+32400+%29%29%2F2%5C-4+=+25

Quadratic expression -4a%5E2%2B20a%2B2000 can be factored:
-4a%5E2%2B20a%2B2000+=+-4%28a--20%29%2A%28a-25%29
Again, the answer is: -20, 25. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+-4%2Ax%5E2%2B20%2Ax%2B2000+%29

a=$25 The original assessment was $25
x=$400/$25=16 There were originally 16 members.
CHECK
ax=(x+4)(a-$5)=$400
$25(16)=20($20)=$400
$400=$400=$400