+1206 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.
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
