I find to find the value of k in terms of $\alpha$ and $\beta$ .
I rearranged the equation $x^2
+kx-1=2k+x$ into $x^2+(k-1)x-(1+2k)=0.$
I then found $\alpha$$\beta$ to equal$-1-2k$.
and $\alpha + \beta$ to equal $-k+1.$
I was then asked to find $k$ for which the equation has equal roots.
I am unsure of what to do.
Does this just mean when $\alpha$ = $\beta$ ? And how do I incorporate my previously found information into it?
Thank you, all help is appreciated.
Best Answer
If you don't want to incorporate your prior knowledge to the problem, the quadratic equation has equal roots when the discriminant is equal to zero. $$(k-1)^2 + 4(1)(1+2k) = k^2 + 6k + 5 = (k+1)(k+5) = 0 \iff k \in \{ -5, -1 \} $$