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 Find the largest integer k such that the equation 5x^2 - kx + 8 - 4x^2 + 10x + 43 = 0 has no real solutions.

 Feb 18, 2024
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\( 5x^2 - kx + 8 - 4x^2 + 10x + 43 = 0\\ x^2+(10-k)x+51=0\)
If the discriminat of a quad eq is less than zero it has no real solutions, therefore:
\(b^2-4ac<0\\ (10-k)^2-4\cdot51<0\\ (10-k)^2<204\)
k must be an integer therefore:
Let \(10-k=a\) Then:
\(a^2<204\\ a<\sqrt{204}\approx\pm14(\text{actual value is 14.28})\\ 10-k=14\\ k=-4\\ 10-k=-14\\ k=24 \)

Therfore k is in the interval \(24\le k\le-4\) Thus the greatest interger that k can be is :
\(\LARGE \boxed{k=24}\)

 Feb 18, 2024

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