Find the set of values of k such that the expression 4x(x + k) − k(3k − 4) is positive for all x.

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pramodpandey | College Teacher | (Level 3) Valedictorian

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We have given

`4x(x+k)-k(3k-4)`

`=4x^2+4xk-3k^2+4k`

`=(4x^2+4xk+k^2)-k^2-3k^2+4k`

`=(2x+k)^2-4k(k-1)`

Now if

`(2x+k)^2-4k(k-1)>0 `

`(2x+k)^2-4k(k-1)>0`

`i.e (2x+k)^2 >0`  always true.

We need only second term to be positive ie. `-4k(k-1)>=0`

either   `k>=0 and k-1<=0` 

ie. `0<=k<=1`          (i)

or   `k<0 and k-1>0`

`k<0 and k>1` 

This case is not possible.

Thus only solution is

`0<=k<=1`  

======================================

In above answer ,let  x=1 ,k=-2

`(x+k/2)^2+k-k^2=(1-1)^2-2-4=-6 <0`

Thus above answer is not correct.

answer does not consider k=0,1

even it can verify that  k=0,1 satisfy the condition.

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