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

magenta Jul 11, 2024

#1**+1 **

Simplify to... 3x^2 - kx + 33 = 0

Then in Desmos, graph

3x^2 - kx + 33 = 0

and

k = 0

Click on k = 0 and set the **Step** to 1

Change the range from "-10 to 10" to "0 to 25"

let the slider play from 25 to 0 and find the first one that has no line shown on it

threepointonefourone Jul 11, 2024

#2**+1 **

First, let's simplify and combine all like terms.

After doing that, we get

\(3x^2-kx+33=0\)

Now, in order for the equation to have no real solutions, the descriminant must be less than 0.

The descriminant is \(b^2-4ac\), so we have the equation

\(k^2 - 4(33)(3) < 0\)

Since we are trying to find the largest, we set the two to equal each other and get

\(k^2 - 396 = 0\\ k^2 = 396\\ k \approx 19.89974\)

Rounding down, the largest k can be as an integer is 19.

So our answer is 19.

Thanks! :)

NotThatSmart Jul 11, 2024