The quadratic equation $7x^2-(k+13)x+k^2-k-2=0$, where $k$ is a constant, has two distinct real roots $\alpha$ and $\beta$. Given that $0<\alpha<1$ and $1<\beta<2,$ find the range of values of k.
I got $−2.055<k<4.055,$ but is unable to link to the alpha and beta.
Best Answer
Let $f(x)= 7x^2-(k+13)x+k^2-k-2.$
Now, solve the following system. $$f(0)>0,$$ $$f(1)<0$$ and $$f(2)>0,$$ which is $$k^2-k-2>0$$ $$k^2-2k-8<0$$ and $$k^2-3k>0$$ Can you end it now?
I got $$(-2,-1)\cup(3,4)$$