$\begin{array} {l} x = \dfrac { 3 + \sqrt{ 39 } } { 3 } \\ x = \dfrac { 3 - \sqrt{ 39 } } { 3 } \end{array}$
$\color{#FF6800}{ 0.3 } x ^ { 2 } - 0.6 x = 1$
$ $ Convert decimals to fractions $ $
$\color{#FF6800}{ \dfrac { \color{#FF6800}{ 3 } } { \color{#FF6800}{ 10 } } } x ^ { 2 } - 0.6 x = 1$
$\dfrac { 3 } { 10 } x ^ { 2 } \color{#FF6800}{ - } \color{#FF6800}{ 0.6 } x = 1$
$ $ Convert decimals to fractions $ $
$\dfrac { 3 } { 10 } x ^ { 2 } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { \color{#FF6800}{ 3 } } { \color{#FF6800}{ 5 } } } x = 1$
$\color{#FF6800}{ \dfrac { \color{#FF6800}{ 3 } } { \color{#FF6800}{ 10 } } } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { \color{#FF6800}{ 3 } } { \color{#FF6800}{ 5 } } } \color{#FF6800}{ x } = 1$
$ $ Calculate the expression as a fraction format $ $
$\color{#FF6800}{ \dfrac { \color{#FF6800}{ 3 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 6 } \color{#FF6800}{ x } } { \color{#FF6800}{ 10 } } } = 1$
$\color{#FF6800}{ \dfrac { \color{#FF6800}{ 3 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 6 } \color{#FF6800}{ x } } { \color{#FF6800}{ 10 } } } = \color{#FF6800}{ 1 }$
$ $ Multiply both sides by the least common multiple for the denominators to eliminate the fraction $ $
$\color{#FF6800}{ 3 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 6 } \color{#FF6800}{ x } = \color{#FF6800}{ 10 }$
$3 x ^ { 2 } - 6 x = \color{#FF6800}{ 10 }$
$ $ Move the expression to the left side and change the symbol $ $
$3 x ^ { 2 } - 6 x \color{#FF6800}{ - } \color{#FF6800}{ 10 } = 0$
$\color{#FF6800}{ 3 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 6 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 10 } = \color{#FF6800}{ 0 }$
$ $ Solve the quadratic equation $ ax^{2}+bx+c=0 $ using the quadratic formula $ \dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ - } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 6 } \right ) \pm \sqrt{ \left ( \color{#FF6800}{ - } \color{#FF6800}{ 6 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 10 } \right ) } } { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } } }$
$x = \dfrac { \color{#FF6800}{ - } \left ( \color{#FF6800}{ - } 6 \right ) \pm \sqrt{ \left ( - 6 \right ) ^ { 2 } - 4 \times 3 \times \left ( - 10 \right ) } } { 2 \times 3 }$
$ $ Simplify Minus $ $
$x = \dfrac { 6 \pm \sqrt{ \left ( - 6 \right ) ^ { 2 } - 4 \times 3 \times \left ( - 10 \right ) } } { 2 \times 3 }$
$x = \dfrac { 6 \pm \sqrt{ \left ( \color{#FF6800}{ - } \color{#FF6800}{ 6 } \right ) ^ { \color{#FF6800}{ 2 } } - 4 \times 3 \times \left ( - 10 \right ) } } { 2 \times 3 }$
$ $ Remove negative signs because negative numbers raised to even powers are positive $ $
$x = \dfrac { 6 \pm \sqrt{ 6 ^ { 2 } - 4 \times 3 \times \left ( - 10 \right ) } } { 2 \times 3 }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 6 } \pm \sqrt{ \color{#FF6800}{ 6 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 10 } \right ) } } { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } } }$
$ $ Organize the expression $ $
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 6 } \pm \sqrt{ \color{#FF6800}{ 156 } } } { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } } }$
$x = \dfrac { 6 \pm \sqrt{ \color{#FF6800}{ 156 } } } { 2 \times 3 }$
$ $ Organize the part that can be taken out of the radical sign inside the square root symbol $ $
$x = \dfrac { 6 \pm \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 39 } } } { 2 \times 3 }$
$x = \dfrac { 6 \pm 2 \sqrt{ 39 } } { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } }$
$ $ Multiply $ 2 $ and $ 3$
$x = \dfrac { 6 \pm 2 \sqrt{ 39 } } { \color{#FF6800}{ 6 } }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 6 } \pm \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 39 } } } { \color{#FF6800}{ 6 } } }$
$ $ Separate the answer $ $
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 6 } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 39 } } } { \color{#FF6800}{ 6 } } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 6 } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 39 } } } { \color{#FF6800}{ 6 } } } \end{array}$
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 6 } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 39 } } } { \color{#FF6800}{ 6 } } } \\ x = \dfrac { 6 - 2 \sqrt{ 39 } } { 6 } \end{array}$
$ $ Do the reduction of the fraction format $ $
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 3 } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 39 } } } { \color{#FF6800}{ 3 } } } \\ x = \dfrac { 6 - 2 \sqrt{ 39 } } { 6 } \end{array}$
$\begin{array} {l} x = \dfrac { 3 + \sqrt{ 39 } } { 3 } \\ x = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 6 } \color{#FF6800}{ - } \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 39 } } } { \color{#FF6800}{ 6 } } } \end{array}$
$ $ Do the reduction of the fraction format $ $
$\begin{array} {l} x = \dfrac { 3 + \sqrt{ 39 } } { 3 } \\ x = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 3 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 39 } } } { \color{#FF6800}{ 3 } } } \end{array}$