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

LiIIiam0216 Sep 7, 2024

#1**+2 **

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.**

mathrokus Sep 7, 2024