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.
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.