$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } \right ) ^ { \color{#FF6800}{ 2 } } = 113$
$ $ Organize the expression $ $
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } = 113$
$2 x ^ { 2 } + 2 x + 1 = \color{#FF6800}{ 113 }$
$ $ Move the expression to the left side and change the symbol $ $
$2 x ^ { 2 } + 2 x + 1 \color{#FF6800}{ - } \color{#FF6800}{ 113 } = 0$
$2 x ^ { 2 } + 2 x + \color{#FF6800}{ 1 } \color{#FF6800}{ - } \color{#FF6800}{ 113 } = 0$
$ $ Subtract $ 113 $ from $ 1$
$2 x ^ { 2 } + 2 x \color{#FF6800}{ - } \color{#FF6800}{ 112 } = 0$
$\color{#FF6800}{ 2 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 112 } = 0$
$ $ Bind the expressions with the common factor $ 2$
$\color{#FF6800}{ 2 } \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 56 } \right ) = 0$
$\color{#FF6800}{ 2 } \left ( \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 56 } \right ) = \color{#FF6800}{ 0 }$
$ $ Divide both sides by $ 2$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 56 } = \color{#FF6800}{ 0 }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 1 \pm \sqrt{ 1 ^ { 2 } - 4 \times 1 \times \left ( - 56 \right ) } } { 2 \times 1 } }$
$ $ Organize the expression $ $
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 1 \pm \sqrt{ 225 } } { 2 \times 1 } }$
$x = \dfrac { - 1 \pm \sqrt{ \color{#FF6800}{ 225 } } } { 2 \times 1 }$
$ $ Organize the part that can be taken out of the radical sign inside the square root symbol $ $
$x = \dfrac { - 1 \pm \color{#FF6800}{ 15 } } { 2 \times 1 }$
$x = \dfrac { - 1 \pm 15 } { 2 \color{#FF6800}{ \times } \color{#FF6800}{ 1 } }$
$ $ Multiplying any number by 1 does not change the value $ $
$x = \dfrac { - 1 \pm 15 } { \color{#FF6800}{ 2 } }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 1 \pm 15 } { 2 } }$
$ $ Separate the answer $ $
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 1 + 15 } { 2 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 1 - 15 } { 2 } } \end{array}$
$\begin{array} {l} x = \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 1 } \color{#FF6800}{ + } \color{#FF6800}{ 15 } } { 2 } \\ x = \dfrac { - 1 - 15 } { 2 } \end{array}$
$ $ Add $ - 1 $ and $ 15$
$\begin{array} {l} x = \dfrac { \color{#FF6800}{ 14 } } { 2 } \\ x = \dfrac { - 1 - 15 } { 2 } \end{array}$
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { 14 } { 2 } } \\ x = \dfrac { - 1 - 15 } { 2 } \end{array}$
$ $ Do the reduction of the fraction format $ $
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { 7 } { 1 } } \\ x = \dfrac { - 1 - 15 } { 2 } \end{array}$
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { 7 } { 1 } } \\ x = \dfrac { - 1 - 15 } { 2 } \end{array}$
$ $ Reduce the fraction to the lowest term $ $
$\begin{array} {l} x = \color{#FF6800}{ 7 } \\ x = \dfrac { - 1 - 15 } { 2 } \end{array}$
$\begin{array} {l} x = 7 \\ x = \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 1 } \color{#FF6800}{ - } \color{#FF6800}{ 15 } } { 2 } \end{array}$
$ $ Find the sum of the negative numbers $ $
$\begin{array} {l} x = 7 \\ x = \dfrac { \color{#FF6800}{ - } \color{#FF6800}{ 16 } } { 2 } \end{array}$
$\begin{array} {l} x = 7 \\ x = \color{#FF6800}{ \dfrac { - 16 } { 2 } } \end{array}$
$ $ Do the reduction of the fraction format $ $
$\begin{array} {l} x = 7 \\ x = \color{#FF6800}{ \dfrac { - 8 } { 1 } } \end{array}$
$\begin{array} {l} x = 7 \\ x = \dfrac { - 8 } { \color{#FF6800}{ 1 } } \end{array}$
$ $ If the denominator is 1, the denominator can be removed $ $
$\begin{array} {l} x = 7 \\ x = \color{#FF6800}{ - } \color{#FF6800}{ 8 } \end{array}$