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# HEELLP

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

Feb 14, 2018

#1
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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

Feb 15, 2018
#2
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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   Feb 15, 2018