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Find the largest integer k such that the equation 5x^2 - kx + 8 - 2x^2 + 25 =0 has no real solutions

 Oct 14, 2024
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\(5x^2 - kx + 8 - 2x^2 + 25 = 0 \\ 3x^2 - kx + 33 = 0\)

 

After some simplification, I arrive at the following quadratic. In order to analyze when this quadratic has no real solutions, evaluate the discriminant, specifically when the discriminant is less than zero.

 

\(\Delta = b^2 - 4ac \\ \Delta = (-k)^2 - 4 * 3 * 33 \\ k^2 - 396 < 0 \\ k^2 < 396 \\ k < \sqrt{396} \approx 19.900\)

 

When the discriminant is less than zero, there are no real solutions, so k = 19 is the largest integer that makes the resulting quadratic have no real solutions.

 Oct 16, 2024

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