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The value K has the property that \(x^2 - 2(4K-1)x + 15K^2 - 2K- 7> 0 \) for all real values of x.  Find all possible values of K.

 Jul 8, 2020
 #1
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We consider the discriminant of the left hand side.

 

Because the quadratic function is always positive, the discriminant is negative.

 

\((2(4K - 1))^2 - 4(15K^2 - 2K - 7) < 0\\ 64K^2 - 32K + 4 - 60K^2 + 8K + 28 < 0\\ 4K^2 - 24K + 32 < 0\\ K^2 - 6K + 8 < 0\\ \)

Solving gives \(2 < K < 4\). I will leave the solving process to you.

 Jul 8, 2020

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