5x^2-8kx+4k^2-4

[Juri]

Hello there! I would like to ask the step-by-step procedure on finding the k value with (5x^2-8kx+4k^2-4) as the equation. I am stuck with this problem The answer is + or - square root of 5, but I don’t know how it ended up with that answer. Your help will mean a lot to me!

 

[Steven G]

You failed to write a question.
Exactly what do you need to do (5x^2-8kx+4k^2-4)?
Do you want to factor (5x^2-8kx+4k^2-4) , do you want to solve (5x^2-8kx+4k^2-4) =0, ...?
Please let us know.

 

[Juri]

Thank you very much for responding! The question translates “find the k value where the ellipse (x^2+4y^2) and the line (y = -x+k) touches or comes into contact with each other.

I want to solve (5x^2-8kx+4k^2-4) = 0 until the answer ends up with

k = (+) or (-) √5

 

[Juri]

It also states that “in order for the line (y = -x+k) and the ellipse (x^2+4y^2=4) to come into contact with each other, the discriminant of the equation 5x^2-8kx+4k^2-4 = 0 has to be set as D, therefore, D = 0.

 

[Dr.Peterson]

So, did you find the discriminant of the equation 5x^2 - 8kx + (4k^2 - 4) = 0 ?

Set that equal to 0 and solve for k; you will get [MATH]\pm\sqrt{5}[/MATH] as the solution.

 

[Juri]

No, I am having a hard time solving for the discriminant as well as k, because this question especially is different from all the other equations I had solved before. If the equation looked something like x^2+3x-8 or x^2+6x+9, I would have been able to solve for the discriminant with ease... I’m hesitant to solve 5x^2-8kx-4k^2-4 because I have no idea how it ended up with (+)(-)√5. I was absent from class when they covered this up in the lecture. Maybe I overanalyzed and forgot something important here. For now, I would like to ask the step-by-step procedure in order to (1.) Find the discriminant of 5x^2-8kx-4k^2-4 and (2.) solve for k afterwards.

Thank you for your time!

 

[Deleted member 4993]

If you have the following quadratic function:

Ax² + Bx + C

The discriminant is:

B² - 4*A*C

Your function is:

5x^2-8kx-4k^2-4

So in your case:

B = -8k and
A =5 and C = -(4k^2 + 4)

so the discriminant in this case is:

Discriminant = 64k^2 - 4*5*[ -(4k^2 + 4)]

continue.....

Do not worry about \(\displaystyle \pm \sqrt{5}\) yet

It will become obvious as you "work" with the discriminant!

 

[Juri]

Thank you very much for this comment!? After so many attempts I was finally able to solve the given problem. Apparently my error was that I failed to apply the basic formula of ax^2+bx+c which made me believe I was going well with

a = 5, b = -8kx (I soon realized the reason why I kept getting wrong answers was because putting the x in there obviously shouldn’t have been done since it is the formula itself), and c = 4 (this was also an error for me because I thought 4k^2 and -4 should be separated, thus, I didn’t know where -4 belonged).

This gave me a relief, I can’t thank you guys enough!)