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If one zero of the polynomial 3x^2 + 12x - k is reciprocal of the other , then find the value of k. ​

 Jun 15, 2022
 #1
avatar+2339 
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Note that if the roots are reciprocals of each other, they will multiply to 1, meaning \(c \over a \) will equal 1, according to Vieta's. 

 

This means \({-c \over 3} = 1\), meaning \(\color{brown}\boxed{k = -3}\)

 Jun 15, 2022
 #2
avatar+124524 
+1

3x^2  + 12x  -  k  =  0

 

Sum of roots  =  -12/ 3  = -4

 

So

 

x + 1/x=  -4

 

x^2 + 4x + 1 =  0

 

x^2 + 4x  =  -1          complete the square on x

 

x^2 + 4x  + 4 =  -1 + 4

 

(x + 2)^2  = 3        take  both roots

 

x + 2 =  sqrt (3)             x + 2 = -sqrt (3)

 

x = sqrt (3) - 2              x =  -sqrt (3) - 2

 

Product of the roots  =   -k /3

 

So

 

(sqrt (3) - 2) (-sqrt (3)  -2)  =  -k/3

 

- (sqrt (3) - 2) (sqrt (3) + 2)  = -k/3

 

(sqrt (3) -2) ( sqrt (3) + 2)  = k /3

 

3 -4  =  k/3

 

-1  = k/3

 

k = -3

 

cool cool cool

 Jun 15, 2022

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