Answer
The solutions are $x = 0, 5 - 2\sqrt {3}, \text{ and } 5 + 2\sqrt {3}$.
Work Step by Step
Rewrite this equation so all the terms are on the left side of the equation and that the equation equals $0$.
$x^3 - 10x^2 + 13x = 0$
Factor out $x$ from each term:
$x(x^2 - 10x + 13) = 0$
Use the Zero-Product Property by equating each factor to $0$, then solve each equation.
First factor:
$x = 0$
Second factor:
$x^2 - 10x + 13 = 0$
We cannot factor this polynomial, so we resort to using the quadratic formula, which is:
$x = \dfrac{-b \pm \sqrt {b^2 - 4ac}}{2a}$
where $a=1, b=-10,$ and $c=13$ is the constant.
Let us plug in the numbers into the formula:
$x = \dfrac{-(-10) \pm \sqrt {(-10)^2 - 4(1)(13)}}{2(1)}$
$x = \dfrac{10 \pm \sqrt {100 - 52}}{2}$
$x = \dfrac{10 \pm \sqrt {48}}{2}$
The number $48$ can be expanded into the factors $16$ and $3$:
$x = \dfrac{10 \pm \sqrt {3(16)}}{2}$
$x = \dfrac{10 \pm 4\sqrt {3}}{2}$
$x = 5 \pm 2\sqrt {3}$
The solutions are $x = 0, 5 \pm 2\sqrt {3}$.
