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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
avatar
+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
 #2
avatar+26973 
+2

Here's an alternative approach:

 

.

Alan  Sep 14, 2018
 #3
avatar+88898 
+2

6 = 35/x  - 49/x^2       mutliply through by x^2

 

6x^2 = 35x   - 49      rearrange as

 

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

 

(3x - 7) (2x - 7)   = 0

 

Set both factors  = 0   and solve for x and we get that    x  = 7/3   or x  = 7/2

 

 

 

cool cool cool

CPhill  Sep 14, 2018

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