Find all values of \(x\) such that \(6= \dfrac{35}{x} -\dfrac{49}{x^2}\). If you find more than one value, then list your solutions in increasing order, separated by commas.

Logic Sep 14, 2018

#1**+2 **

Solve for x:

6 = 35/x - 49/x^2

6 = 35/x - 49/x^2 is equivalent to 35/x - 49/x^2 = 6:

35/x - 49/x^2 = 6

Bring 35/x - 49/x^2 together using the common denominator x^2:

(7 (5 x - 7))/x^2 = 6

Multiply both sides by x^2:

7 (5 x - 7) = 6 x^2

Expand out terms of the left hand side:

35 x - 49 = 6 x^2

Subtract 6 x^2 from both sides:

-6 x^2 + 35 x - 49 = 0

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

-(2 x - 7) (3 x - 7) = 0

Multiply both sides by -1:

(2 x - 7) (3 x - 7) = 0

Split into two equations:

2 x - 7 = 0 or 3 x - 7 = 0

Add 7 to both sides:

2 x = 7 or 3 x - 7 = 0

Divide both sides by 2:

x = 7/2 or 3 x - 7 = 0

Add 7 to both sides:

x = 7/2 or 3 x = 7

Divide both sides by 3:

**x = 7/2 or x = 7/3**

Guest Sep 14, 2018