+2669 Find the largest integer k such that the equation 5x^2 - kx + 8 - 2x^2 + 25 =0 has no real solutions
+23 5x^2 - kx + 8 -2x^2 + 25 = 0
Simplifying,
3x^2 - kx + 33 = 0
a = 3, b = k, c = 33
The formula for the discriminant is b^2 - 4ac.
For the quadratic to have no real solutions, the discriminant must be negative.
So, b^2 - 4ac < 0
Plugging in the values,
k^2 - 4(3)(33) < 0
k^2 -396 < 0
k^2 < 396
For this question, we have to find the largest value of k that is a solution to k^2 < 396.
To maximize k, k should also be positive, (-k)^2 would always have the same solution as k^2
We know that 396 ~ 400, so we can test squares.
20^2 = 400
19 ^ 2 = 361
Therefore, our solution is k = 19.
