Solve the equation by completing the square.
-$$x ^ { 2 } + 2 = - x$$
A) $\left\{ \frac { 1 - \mathrm { i } \sqrt { 7 } } { 2 } , \frac { 1 + \mathrm { i } \sqrt { 7 } } { 2 } \right\}$
B) $\left\{ \frac { - 1 - i \sqrt { 7 } } { 2 } , \frac { - 1 + i \sqrt { 7 } } { 2 } \right\}$
C) $\left\{ \frac { - 1 - \sqrt { 7 } } { 2 } , \frac { - 1 + \sqrt { 7 } } { 2 } \right\}$
D) $\left\{ \frac { 1 - \sqrt { 7 } } { 2 } , \frac { 1 + \sqrt { 7 } } { 2 } \right\}$

Question

Quizplus · Accepted Answer

To solve the equation $x^2 + 2 = -x$ by completing the square, first move all terms to one side to get $x^2 + x + 2 = 0$. Completing the square involves making the left side of the equation a perfect square trinomial. To do this, add and subtract $\left(\frac{1}{2}\right)^2 = \frac{1}{4}$ inside the equation, leading to $x^2 + x + \frac{1}{4} - \frac{1}{4} + 2 = 0$, which simplifies to $\left(x + \frac{1}{2}\right)^2 + \frac{7}{4} = 0$. Solving for $x$ gives $x + \frac{1}{2} = \pm i\sqrt{\frac{7}{4}}$, which simplifies to $x = -\frac{1}{2} \pm i\frac{\sqrt{7}}{2}$, matching choice B: $\left\{ \frac{-1 - i\sqrt{7}}{2}, \frac{-1 + i\sqrt{7}}{2} \right\}$.

Solve the Equation by Completing the Square x2+2=−xx ^ { 2 } + 2 = - xx2+2=−x

Solve the Equation by Completing the Square $x ^ { 2 } + 2 = - x$