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given that a and b are the roots a quadratic equation 2x2-2x+4 find a quadratic equation whose roots are a/b and b/a

Guest Nov 18, 2018
edited by Guest  Nov 18, 2018
 #1
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2x^2 -2x + 4 = 0       divide through by 2

x^2 - x + 2  = 0        subtract 2 from both sides

x^2 - x  =  - 2       complete the square on x

x^2 - x + 1/4  =  -2 + 1/4

(x - 1/2)^2  = -7/4       take both roots

x - 1/2  = ±i√7 / 2        add 1/2 to both sides

So

x = [ i√7 + 1] / 2     =  a

a^2  =  [ 1 + √7 i] [ 1 + √7 i ] / 4  =  [ 1 + 2√7 i - 7] / 4  =  [ √7 i - 3 ] / 2

Or

x = [ -i√7 + 1] / 2  =    b

b^2 =  [ 1 - √7 i ]  [ 1 - √7 i ] / 4 =  [ 1 -2√7 i  -  7 ] / 4  =  [ -√7 i - 3] / 2

 

a^2 + b^2   =   -6/ 2  =  -3

 

ab  =   [  1 +  i√7 ] / 2  *  [  1  - i√7 ] / 2   =  [ 1 + 7] / 4  =  2

 

 

So    ( x - a/b)  (x - b/a)  =

 

x^2 - (a/b)x   - (b/a)x +  (a/b)(b/a)  =

 

x^2 - ( a/b + b/a)x + 1   =

 

x^2 - ( a^2 + b^2) x / ab + 1  =

 

x^2 -  (-3)x / 2 + 1   =

 

x^2 + (3/2)x + 1

 

 

 

cool cool cool

CPhill  Nov 18, 2018

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