For which value of $k$ does the equation $x^4 – 4x^2 + x + k = 0 $ have four distinct real roots?
I found this question on a standardized test, and the answer presumably relies on a graphing calculator. Is there some method to solve this problem without the aid of a calculator?
Best Answer
Discriminant of equation is $$ \Delta(k) = \begin{array}{|ccccccc|} 1 & 0 & -4 & 1 & k & 0 & 0 \\ 0 & 1 & 0 & -4 & 1 & k & 0 \\ 0 & 0 & 1 & 0 & -4 & 1 & k \\ 4 & 0 & -8 & 1 & 0 & 0 & 0 \\ 0 & 4 & 0 & -8 & 1 & 0 & 0 \\ 0 & 0 & 4 & 0 & -8 & 1 & 0 \\ 0 & 0 & 0 & 4 & 0 & -8 & 1 \end{array} = 256 k^3 - 2048 k^2 + 3520 k + 229. $$
If equation has distinct real roots, then discriminant is positive.
I hope we are talking about $k\in \mathbb{Z}$.
If $k\le -1$, then $\Delta(k)<0$.
$\Delta(0)=229>0$.
$\Delta(1)=1957>0$.
$\Delta(2)=1125>0$.
$\Delta(3)=-731<0$.
$\Delta(4)=-2075<0$.
$\Delta(5)=-1371<0$.
If $k\ge 6$, then $\Delta(k)>0$.
When $k\ge 6$, then $x^4 - 4x^2 + x + k>0$.
So, $k =0$, $k=1$, or $k=2$.
Links to see graphs/plots (and related info):
$x^4-4x^2+x+0$,
$x^4-4x^2+x+1$,
$x^4-4x^2+x+2$.