Find the value of $k$ so that the two roots of a quadratic equation are between $-2$ and $4$

Question

I need help with this:

In what interval does $k$ need to vary so that the two roots of the equation

$$x^2 - 2kx + k^2-1 = 0$$

are between $-2$ and $4$?

I think the answer is $k = [-1, 3]$, but I found that out by trial and error. How could I solve this problem? Thanks in advance.

Answer 1

$$x^2-2kx+k^2-1=0$$

$$\Leftrightarrow (x-k)^2-1=0$$

$$\Leftrightarrow (x-k-1)(x-k+1)=0$$

$$\Leftrightarrow x_{1}=k-1;x_{2}=k+1$$.

We have $-2 \le x_{1},x_{2} \le 4$ and because $x_{1}<x_{2}$ for all $k$, we will have $-2 \le k-1<k+1 \le 4$ or ${\begin{cases}-2\le k-1\\k+1\le 4\end{cases}}\Rightarrow {\begin{cases}-1\le k\\k\le 3\end{cases}}\Rightarrow -1\le k\le 3$.

Answer 2

The two root of the equation is $x=k\pm 1$, so you have to solve the inequality $$ \begin{cases} -2 \le k+1 \le 4\\ -2 \le k-1 \le 4 \end{cases}\Rightarrow -2 \le k-1\le k+1 \le 4. $$ Thus $-1\le k \le 3$.