I used to learn this but I forgot all the stuff regarding this

[Cat]

α and β are roots of the above equation and 1/α+1/β=-1/3
Part a asks the value of k and part b asks the value of α^3+4β^2. A little help with elaborate steps and explanation will be nice. Thank you.

 

[stapel]

A little help with elaborate steps and explanation will be nice.

Yes, fully-worked solutions that one can simply copy into one's homework can be nice, at least in the short term. But since the forum's helpers won't be available to fill in your tests for you, it wouldn't be nice in the long term. Also, it could be viewed as cheating. :shock:

27. \(\displaystyle x^2\, +\, (k\, +\, 2)x\, +\, k\, =\, 0,\) where \(\displaystyle \alpha\) and \(\displaystyle \beta\) are the roots of the equation, and \(\displaystyle \frac{1}{\alpha}\, +\, \frac{1}{\beta}\, =\, -\frac{1}{3}\)
(a) Find the value of \(\displaystyle k\)
(b) Find the value of \(\displaystyle \alpha^3\, +\, 4\beta^2\)

Part a asks the value of k and part b asks the value of α^2+4β^3.

a. What did you get when you plugged the equation into the Quadratic Formula? What did the Formula give you for the stuff inside the square root? Note: For this equation to have real-number solutions, the expression inside the square root has to be non-negative. What does this tell you?

 

[Cat]

Yes, fully-worked solutions that one can simply copy into one's homework can be nice, at least in the short term. But since the forum's helpers won't be available to fill in your tests for you, it wouldn't be nice in the long term. Also, it could be viewed as cheating. :shock:


a. What did you get when you plugged the equation into the Quadratic Formula? What did the Formula give you for the stuff inside the square root? Note: For this equation to have real-number solutions, the expression inside the square root has to be non-negative. What does this tell you?

In fact, it's not me who needed help. I was asked by my friend regarding this question. But then I realised he, as well as me, didn't study this anymore, so maybe he was asking this for his friends/young siblings.

And yes I have studied the Quadratic formula and I knew what you meant, but I still couldn't figure out the answer. I wish I could go to high school and re-study all that, but I am not in high school anymore and what I am studying now is completely irrelevant to this, and therefore even if you explain it in details, you shouldn't consider this as cheating.

 

[HallsofIvy]

If \(\displaystyle \alpha\) and \(\displaystyle \beta\) are roots of \(\displaystyle x^2+ (k+ 2)x+ k= 0\)
then \(\displaystyle (x- \alpha)(x- \beta)= x^2- (\alpha+ \beta)x+ \alpha\beta= x^2+ (k+ 2)x+ k\)
for all x so we must have \(\displaystyle \alpha+ \beta= -k- 2\) and \(\displaystyle \alpha\beta= k\).

Multiplying both sides of \(\displaystyle \frac{1}{\alpha}+ \frac{1}{\beta}= -\frac{1}{3}\) by \(\displaystyle -3\alpha\beta\),
\(\displaystyle -3\beta- 3\alpha= \alpha\beta\) so that \(\displaystyle \alpha+ \beta= -\frac{\alpha\beta}{3}\).

So \(\displaystyle \alpha+ \beta= k+ 2= -\frac{k}{3}\). Solve that for k.