Answer
The solutions are $x = 10, -5 - 5i\sqrt {3}, \text{ and } -5 - 5i\sqrt {3}$.
Work Step by Step
We see that $x^3 - 1000$ is the difference of two cubes.
We can factor using the formula:
$(a - b)(a^2 + ab + b^2)$
We plug in the values, where $a = \sqrt[3] {x^3}$ (or $a = x$) and $b = \sqrt[3] {1000}$ (or $b = 10$:):
$(x - 10)(x^2 + 10x + 10^2) = 0$
$(x - 10)(x^2 + 10x + 100) = 0$
We set each factor to $0$, then solve each equation.
First factor:
$x - 10 = 0$
$x = 10$
Second factor:
$x^2 + 10x + 100 = 0$
We cannot factor this polynomial, so we resort to using the quadratic formula, which is:
$x = \dfrac{-b ± \sqrt {b^2 - 4ac}}{2a}$
where $a$ is the coefficient of the first term, $b$ is the coefficient of the 1st degree term, and $c$ is the constant.
Substitute $a=1, b=10,$ and $c=100$ into the formula:
$x = \dfrac{-10) ± \sqrt {(10)^2 - 4(1)(100)}}{2(1)}$
$x = \dfrac{-10) ± \sqrt {100 - 400}}{2}$
$x = \dfrac{-10) ± \sqrt {-300}}{2}$
$x = \dfrac{-10 ± \sqrt {3(-100)}}{2}$
We can take out $-100$ from the radical because the square root of $-100$ is $10i$:
$x = \dfrac{-10 ± 10i\sqrt {3}}{2}$
$x = -5 ± 5i\sqrt {3}$
The solutions are $x = 10, -5 - 5i\sqrt {3}, \text{ and } -5 - 5i\sqrt {3}$.
