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Formula
$$x ^ { 2 } + x + 1 = 0$$
$\begin{array} {l} x = \dfrac { - 1 + \sqrt{ 3 } i } { 2 } \\ x = \dfrac { - 1 - \sqrt{ 3 } i } { 2 } \end{array}$
Calculate using the quodratic formula$($Imaginary root solution$)$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } = \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 { - 1 \pm \sqrt{ 1 ^ { 2 } - 4 \times 1 \times 1 } } { 2 \times 1 } }x = \dfrac { - 1 \pm \sqrt{ \color{#FF6800}{ 1 } ^ { \color{#FF6800}{ 2 } } - 4 \times 1 \times 1 } } { 2 \times 1 } $Calculate power$ x = \dfrac { - 1 \pm \sqrt{ \color{#FF6800}{ 1 } - 4 \times 1 \times 1 } } { 2 \times 1 }x = \dfrac { - 1 \pm \sqrt{ 1 - 4 \color{#FF6800}{ \times } \color{#FF6800}{ 1 } \color{#FF6800}{ \times } \color{#FF6800}{ 1 } } } { 2 \times 1 } $Multiplying any number by 1 does not change the value$ x = \dfrac { - 1 \pm \sqrt{ 1 - 4 } } { 2 \times 1 }x = \dfrac { - 1 \pm \sqrt{ \color{#FF6800}{ 1 } \color{#FF6800}{ - } \color{#FF6800}{ 4 } } } { 2 \times 1 } $Subtract$ 4 $from$ 1x = \dfrac { - 1 \pm \sqrt{ \color{#FF6800}{ - } \color{#FF6800}{ 3 } } } { 2 \times 1 }x = \dfrac { - 1 \pm \sqrt{ - 3 } } { 2 \color{#FF6800}{ \times } \color{#FF6800}{ 1 } } $Multiplying any number by 1 does not change the value$ x = \dfrac { - 1 \pm \sqrt{ - 3 } } { \color{#FF6800}{ 2 } }x = \dfrac { - 1 \pm \sqrt{ \color{#FF6800}{ - } 3 } } { 2 } $Subtracting (-) from the square root gives i$ x = \dfrac { - 1 \pm \sqrt{ 3 } \color{#FF6800}{ i } } { 2 }\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 1 \pm \sqrt{ 3 } i } { 2 } } $Separate the answer$ \begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 1 + \sqrt{ 3 } i } { 2 } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { - 1 - \sqrt{ 3 } i } { 2 } } \end{array}\$