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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? : 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(1156) 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(1+x)
You then add 3500 and let the sum accumulate interest for another year.
(10000(1+x) + 3500)(1+x)
At the end of the year you are told you have 15569.75
So
(10000+10000x + 3500)(1+x) = 15569.75
13500 + 10000x)(1+x) = 15569.75
13500 + 10000x + 13500x + 10000x^2 = 15569.75
10000x^2 + 23500x - 2069.75 = 0
use the quadratic equation to solve for x
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax^2+bx+c=0 (in our case 10000x^2+23500x+-2069.75 = 0) has the following solutons:

x[12] = (b+-sqrt( b^2-4ac ))/2\a

For these solutions to exist, the discriminant b^2-4ac should not be a negative number.

First, we need to compute the discriminant b^2-4ac: b^2-4ac=(23500)^2-4*10000*-2069.75=635040000.

Discriminant d=635040000 is greater than zero. That means that there are two solutions: x[12] = (-23500+-sqrt( 635040000 ))/2\a.

x[1] = (-(23500)+sqrt( 635040000 ))/2\10000 = 0.085
x[2] = (-(23500)-sqrt( 635040000 ))/2\10000 = -2.435

Quadratic expression 10000x^2+23500x+-2069.75 can be factored:
10000x+23500x+-2069.75 = 10000(x-0.085)*(x--2.435)
Again, the answer is: 0.085, -2.435. Here's your graph:
graph( 500, 500, -10, 10, -20, 20, 10000*x^2+23500*x+-2069.75 )


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