Answer
The solutions are $x = 3 - 2\sqrt {2}$ and $x = 3 + 2\sqrt {2}$.
Work Step by Step
We are asked to use the quadratic formula to factor this polynomial. The quadratic formula is given by:
$x = \dfrac{-b \pm \sqrt {b^2 - 4ac}}{2a}$
where $a$ is the coefficient of the $x^2$ term, $b$ is the coefficient of the $x$ term, and $c$ is the constant.
We have $a=1, b=-6,$ and $c=1$ so substituting them into the formula gives::
$x = \dfrac{-(-6) \pm \sqrt {(-6)^2 - 4(1)(1)}}{2(1)}$
$x = \dfrac{6 \pm \sqrt {36 - 4}}{2}$
$x = \dfrac{6 \pm \sqrt {32}}{2}$
The number $32$ can be factored as $16(2)$ so:
$x = \dfrac{6 \pm \sqrt {(16)(2)}}{2}$
We bring out the square root of $16$ which is $4$, so we have:
$x = \dfrac{6 \pm 4\sqrt {2}}{2}$
Divide both numerator and denominator by $2$ to obtain:
$x = 3 \pm 2\sqrt {2}$
Thus, the solutions are $x = 3 ± 2\sqrt {2}$.
