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Find all values of x that satisfy the equation\( \frac {12x}{x^2 + 8} = 2. \)\(Let $s$ and $t$ be the solutions of the quadratic $4x^2 + 9x - 6 = 0.$ Find $$\frac st + \frac ts.$$\)
\(Let $a$ and $b$ be the solutions of the quadratic equation $2x^2 - 8x + 7 = 0$. Find \[\frac{1}{2a} + \frac{1}{2b}.\]\)

Guest Feb 14, 2018
 #1
avatar+13014 
+1

Multiply both sides by x^2 +8    to get

 

12x = 2x^2 + 16     re-arrange

0 = 2x^2 -12x + 16   Divide both sides by two

0=x^2 - 6x + 8        Factor

(x-4)(x-2) = 0

 

For this to be true  either   x-4 = 0    or   x-2 = 0     so x=   4  or 2

ElectricPavlov  Feb 15, 2018
 #2
avatar+89803 
+1

4x^2 + 9x -  6  =  0

 

s/t  + t/s  =      [ s^2 + t^2 ]  [  st ]

 

Using the quadratic function to solve

 

x  =   ( -9 ± √ [ (9^2  - 4* 4 *  - 6) ]  ) /  8

x   =  ( -9 ± √ [ 177]  )/ 8

 

Call  s  =     ( -9 + √ [ 177]  )/ 8        Call t  =   ( -9 - √ [ 177]  )/ 8

 

s^2  = [ 81 - 18√ 177 + 177] / 64      =   [258  - 18√177 ] / 64

 

t^2  =  [  81  + 18√177 + 177 ] / 64  =  [ 258+ 18√177] / 64

 

s^2 +  t^2   =     516 / 64  =   129/16

 

And  st  will   = c / a  =   -6/4  =  -3/2

 

So

 

[ s^2 + t^2 ]  /  st   =   129/16  *  -2 / 3   =   - 43 / 8

 

 

cool cool cool

CPhill  Feb 15, 2018

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