SOLUTION: Could anyone help me explain set-up of this equation: A business invests $10,000 in a savings account for two years. At the beginning of the second year, an additional $3500 is

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Question 149764: Could anyone help me explain set-up of this equation:
A business invests $10,000 in a savings account for two years. At the beginning of the second year, an additional $3500 is invested. At the end of the second year, the account balance is $15,569.75. What was the annual interest rate?
Answer by vleith(2983) About Me  (Show Source):
You can put this solution on YOUR website!
Let x be the interest rate. For example if x = 9% , then use x =0.09
You invest 10000 at rate x. After one year you will have 10000%281%2Bx%29
You then add 3500 and let the sum accumulate interest for another year.
%2810000%281%2Bx%29+%2B+3500%29%281%2Bx%29+
At the end of the year you are told you have 15569.75
So
%2810000%2B10000x+%2B+3500%29%281%2Bx%29+=+15569.75
13500+%2B+10000x%29%281%2Bx%29+=+15569.75
13500+%2B+10000x+%2B+13500x+%2B+10000x%5E2+=+15569.75
10000x%5E2+%2B+23500x+-+2069.75+=+0
use the quadratic equation to solve for x
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 10000x%5E2%2B23500x%2B-2069.75+=+0) has the following solutons:

x%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=%2823500%29%5E2-4%2A10000%2A-2069.75=635040000.

Discriminant d=635040000 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-23500%2B-sqrt%28+635040000+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%2823500%29%2Bsqrt%28+635040000+%29%29%2F2%5C10000+=+0.085
x%5B2%5D+=+%28-%2823500%29-sqrt%28+635040000+%29%29%2F2%5C10000+=+-2.435

Quadratic expression 10000x%5E2%2B23500x%2B-2069.75 can be factored:
10000x%5E2%2B23500x%2B-2069.75+=+10000%28x-0.085%29%2A%28x--2.435%29
Again, the answer is: 0.085, -2.435. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+10000%2Ax%5E2%2B23500%2Ax%2B-2069.75+%29


The end amount is greater than the amount you put in, so the interest rate must be positive.
The positive answer is 0.085. Which results in a return of 8.5%.
Check your answer by plugging in 0.085 and verifying the result