Answer
Zeros: $2,-2,\sqrt{6},-\sqrt{6} $
$x$-intercepts: $2,-2,\sqrt{6},-\sqrt{6} $
Work Step by Step
To find the zeros of a function $f$, solve the equation $f(x)=0$
The zeros of the function are also the $x-$intercepts.
Let $f(x)=0$:
$$x^4-10x^2+24=0$$
Let $u=x^2$, the original equation becomes
$$u^2-10u+24=0$$
By factoring
$$(u-4)(u-6) = 0$$
Use the Zero-Product Property by equating each factor to zero, then solve each equation to obtain:
\begin{align*}
u-4 &= 0 &\text{ or }& &u-6=0\\
u &= 4 &\text{ or }& &u=6\\
\end{align*}
To solve for $x$, we use $u=x^2$
$$\because u = x^2$$
$$\therefore x = \pm \sqrt{u}$$
For $u=4$
$$x=\pm \sqrt{4} $$
$$x= \pm 2$$
For $u=6$
$$x= \pm \sqrt{6}$$
$\therefore x = 2,-2,\sqrt{6},-\sqrt{6} $
