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$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 1 } = \color{#FF6800}{ 0 }$

$ $ Convert the quadratic expression on the left side to a perfect square format $ $

$\left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 1 } \color{#FF6800}{ - } \color{#FF6800}{ 1 } ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 0 }$

$\left ( x + 1 \right ) ^ { 2 } \color{#FF6800}{ - } \color{#FF6800}{ 1 } \color{#FF6800}{ - } \color{#FF6800}{ 1 } ^ { \color{#FF6800}{ 2 } } = 0$

$ $ Move the constant to the right side and change the sign $ $

$\left ( x + 1 \right ) ^ { 2 } = \color{#FF6800}{ 1 } \color{#FF6800}{ + } \color{#FF6800}{ 1 } ^ { \color{#FF6800}{ 2 } }$

$\left ( x + 1 \right ) ^ { 2 } = 1 + \color{#FF6800}{ 1 } ^ { \color{#FF6800}{ 2 } }$

$ $ Calculate power $ $

$\left ( x + 1 \right ) ^ { 2 } = 1 + \color{#FF6800}{ 1 }$

$\left ( x + 1 \right ) ^ { 2 } = \color{#FF6800}{ 1 } \color{#FF6800}{ + } \color{#FF6800}{ 1 }$

$ $ Add $ 1 $ and $ 1$

$\left ( x + 1 \right ) ^ { 2 } = \color{#FF6800}{ 2 }$

$\left ( \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } \right ) ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 2 }$

$ $ Solve quadratic equations using the square root $ $

$\color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } = \pm \sqrt{ \color{#FF6800}{ 2 } }$

$\color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } = \pm \sqrt{ \color{#FF6800}{ 2 } }$

$ $ Solve a solution to $ x$

$\color{#FF6800}{ x } = \pm \sqrt{ \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 1 }$

$\color{#FF6800}{ x } = \pm \sqrt{ \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 1 }$

$ $ Separate the answer $ $

$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ - } \color{#FF6800}{ 1 } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 2 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ - } \color{#FF6800}{ 1 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 2 } } \end{array}$

$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 2 \pm \sqrt{ 2 ^ { 2 } - 4 \times 1 \times \left ( - 1 \right ) } } { 2 \times 1 } }$

$ $ Organize the expression $ $

$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 2 \pm \sqrt{ 8 } } { 2 \times 1 } }$

$x = \dfrac { - 2 \pm \sqrt{ \color{#FF6800}{ 8 } } } { 2 \times 1 }$

$ $ Organize the part that can be taken out of the radical sign inside the square root symbol $ $

$x = \dfrac { - 2 \pm \color{#FF6800}{ 2 } \sqrt{ \color{#FF6800}{ 2 } } } { 2 \times 1 }$

$x = \dfrac { - 2 \pm 2 \sqrt{ 2 } } { 2 \color{#FF6800}{ \times } \color{#FF6800}{ 1 } }$

$ $ Multiplying any number by 1 does not change the value $ $

$x = \dfrac { - 2 \pm 2 \sqrt{ 2 } } { \color{#FF6800}{ 2 } }$

$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 2 \pm 2 \sqrt{ 2 } } { 2 } }$

$ $ Separate the answer $ $

$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 2 + 2 \sqrt{ 2 } } { 2 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 2 - 2 \sqrt{ 2 } } { 2 } } \end{array}$

$\begin{array} {l} x = \color{#FF6800}{ \dfrac { - 2 + 2 \sqrt{ 2 } } { 2 } } \\ x = \dfrac { - 2 - 2 \sqrt{ 2 } } { 2 } \end{array}$

$ $ Do the reduction of the fraction format $ $

$\begin{array} {l} x = \color{#FF6800}{ \dfrac { - 1 + \sqrt{ 2 } } { 1 } } \\ x = \dfrac { - 2 - 2 \sqrt{ 2 } } { 2 } \end{array}$

$\begin{array} {l} x = \dfrac { - 1 + \sqrt{ 2 } } { \color{#FF6800}{ 1 } } \\ x = \dfrac { - 2 - 2 \sqrt{ 2 } } { 2 } \end{array}$

$ $ If the denominator is 1, the denominator can be removed $ $

$\begin{array} {l} x = \color{#FF6800}{ - } \color{#FF6800}{ 1 } \color{#FF6800}{ + } \sqrt{ \color{#FF6800}{ 2 } } \\ x = \dfrac { - 2 - 2 \sqrt{ 2 } } { 2 } \end{array}$

$\begin{array} {l} x = - 1 + \sqrt{ 2 } \\ x = \color{#FF6800}{ \dfrac { - 2 - 2 \sqrt{ 2 } } { 2 } } \end{array}$

$ $ Do the reduction of the fraction format $ $

$\begin{array} {l} x = - 1 + \sqrt{ 2 } \\ x = \color{#FF6800}{ \dfrac { - 1 - \sqrt{ 2 } } { 1 } } \end{array}$

$\begin{array} {l} x = - 1 + \sqrt{ 2 } \\ x = \dfrac { - 1 - \sqrt{ 2 } } { \color{#FF6800}{ 1 } } \end{array}$

$ $ If the denominator is 1, the denominator can be removed $ $

$\begin{array} {l} x = - 1 + \sqrt{ 2 } \\ x = \color{#FF6800}{ - } \color{#FF6800}{ 1 } \color{#FF6800}{ - } \sqrt{ \color{#FF6800}{ 2 } } \end{array}$

$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 1 } = \color{#FF6800}{ 0 }$

$ $ Determine the number of roots using discriminant, $ D=b^{2}-4ac $ from quadratic equation, $ ax^{2}+bx+c=0$

$\color{#FF6800}{ D } = \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 1 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 1 } \right )$

$D = \color{#FF6800}{ 2 } ^ { \color{#FF6800}{ 2 } } - 4 \times 1 \times \left ( - 1 \right )$

$ $ Calculate power $ $

$D = \color{#FF6800}{ 4 } - 4 \times 1 \times \left ( - 1 \right )$

$D = 4 - 4 \color{#FF6800}{ \times } \color{#FF6800}{ 1 } \times \left ( - 1 \right )$

$ $ Multiplying any number by 1 does not change the value $ $

$D = 4 - 4 \times \left ( - 1 \right )$

$D = 4 \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 1 } \right )$

$ $ Multiply $ - 4 $ and $ - 1$

$D = 4 + \color{#FF6800}{ 4 }$

$D = \color{#FF6800}{ 4 } \color{#FF6800}{ + } \color{#FF6800}{ 4 }$

$ $ Add $ 4 $ and $ 4$

$D = \color{#FF6800}{ 8 }$

$\color{#FF6800}{ D } = \color{#FF6800}{ 8 }$

$ $ Since $ D>0 $ , the number of real root of the following quadratic equation is 2 $ $

$ $ 2 real roots $ $

$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 1 } = \color{#FF6800}{ 0 }$

$ $ In the quadratic equation $ ax^{2}+bx+c=0 $ , if the two roots are $ \alpha, \beta $ , then it is $ \alpha + \beta =-\dfrac{b}{a} $ , $ \alpha\times\beta=\dfrac{c}{a}$

$\color{#FF6800}{ \alpha } \color{#FF6800}{ + } \color{#FF6800}{ \beta } = \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 2 } { 1 } } , \color{#FF6800}{ \alpha } \color{#FF6800}{ \beta } = \color{#FF6800}{ \dfrac { - 1 } { 1 } }$

$\alpha + \beta = - \dfrac { 2 } { \color{#FF6800}{ 1 } } , \alpha \beta = \dfrac { - 1 } { 1 }$

$ $ If the denominator is 1, the denominator can be removed $ $

$\alpha + \beta = - \color{#FF6800}{ 2 } , \alpha \beta = \dfrac { - 1 } { 1 }$

$\alpha + \beta = - 2 , \alpha \beta = \dfrac { - 1 } { \color{#FF6800}{ 1 } }$

$ $ If the denominator is 1, the denominator can be removed $ $

$\alpha + \beta = - 2 , \alpha \beta = \color{#FF6800}{ - } \color{#FF6800}{ 1 }$