Suppose the decimal \(0.28\) is equal to \(\frac{1}{n}+\frac{2}{n^2}\), where \(n\) is negative. Find \(n\).

Logic
Sep 14, 2018

#1**+1 **

Solve for n:

1/n + 2/n^2 = 0.28

0.28 = 7/25:

1/n + 2/n^2 = 7/25

Bring 1/n + 2/n^2 together using the common denominator n^2:

(n + 2)/n^2 = 7/25

Cross multiply:

25 (n + 2) = 7 n^2

Expand out terms of the left hand side:

25 n + 50 = 7 n^2

Subtract 7 n^2 from both sides:

-7 n^2 + 25 n + 50 = 0

The left hand side factors into a product with three terms:

-(n - 5) (7 n + 10) = 0

Multiply both sides by -1:

(n - 5) (7 n + 10) = 0

Split into two equations:

n - 5 = 0 or 7 n + 10 = 0

Add 5 to both sides:

n = 5 or 7 n + 10 = 0

Subtract 10 from both sides:

n = 5 or 7 n = -10

Divide both sides by 7:

n = 5 or **n = -10/7**

Guest Sep 15, 2018