Question

what are all real and complex roots of the following function?
0 = x⁴ + 34x² - 72
the real and complex roots of the given function are.
(simplify your answer. use a comma to separate answers as needed. type an exact answer, using radicals and i as needed.)

Explanation:

Step1: Substitute $u=x^2$

Let $u = x^2$, so the equation becomes:
$$u^2 + 34u - 72 = 0$$

Step2: Solve quadratic for $u$

Use quadratic formula $u=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$, where $a=1, b=34, c=-72$:
$$u=\frac{-34\pm\sqrt{34^2-4(1)(-72)}}{2(1)}$$
$$u=\frac{-34\pm\sqrt{1156+288}}{2}$$
$$u=\frac{-34\pm\sqrt{1444}}{2}$$
$$u=\frac{-34\pm38}{2}$$
Calculate two solutions:
$$u_1=\frac{-34+38}{2}=2, \quad u_2=\frac{-34-38}{2}=-36$$

Step3: Solve for $x$ from $u=x^2$

For $u=2$:
$$x^2=2 \implies x=\pm\sqrt{2}$$
For $u=-36$:
$$x^2=-36 \implies x=\pm6i$$

Answer:

$\sqrt{2}, -\sqrt{2}, 6i, -6i$