Identify Equation Type: Identify the type of equation. The given equation x^2 + x + 2 = 0 is a quadratic equation in the standard form ax^2 + bx + c = 0, where a = 1, b = 1, and c = 2.Apply Quadratic Formula: Apply the quadratic formula to find the solutions. The quadratic formula is x = (-b ± √(b^2 - 4ac)) / (2a). For our equation, a = 1, b = 1, and c = 2.Calculate Discriminant: Calculate the discriminant. The discriminant is the part of the quadratic formula under the square root: b^2 - 4ac. Let's calculate it: (1)^2 - 4(1)(2) = 1 - 8 = -7.Determine Root Nature: Determine the nature of the roots. Since the discriminant is negative (-7), the equation has two complex solutions.Calculate Solutions: Calculate the solutions using the quadratic formula. x = (-1 ± √(-7)) / (2*1) = (-1 ± i√7) / 2, where i is the imaginary unit.Write Final Solutions: Write the final solutions. The solutions to the equation x^2 + x + 2 = 0 are x = (-1 + i√7) / 2 and x = (-1 - i√7) / 2.

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x^{2}+x+2=0

Identify Equation Type: Identify the type of equation. $\newline$ The given equation $x^2 + x + 2 = 0$ is a quadratic equation in the standard form $ax^2 + bx + c = 0$ , where $a = 1$ , $b = 1$ , and $c = 2$ .
Apply Quadratic Formula: Apply the quadratic formula to find the solutions. $\newline$ The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . For our equation, $a = 1$ , $b = 1$ , and $c = 2$ .
Calculate Discriminant: Calculate the discriminant. $\newline$ The discriminant is the part of the quadratic formula under the square root: $b^2 - 4ac$ . Let's calculate it: $(1)^2 - 4(1)(2) = 1 - 8 = -7$ .
Determine Root Nature: Determine the nature of the roots. $\newline$ Since the discriminant is negative $-7$ , the equation has two complex solutions.
Calculate Solutions: Calculate the solutions using the quadratic formula. $x = \frac{-1 \pm \sqrt{-7}}{2\cdot1} = \frac{-1 \pm i\sqrt{7}}{2}$ , where $i$ is the imaginary unit.
Write Final Solutions: Write the final solutions. $\newline$ The solutions to the equation $x^2 + x + 2 = 0$ are $x = (-1 + i\sqrt{7}) / 2$ and $x = (-1 - i\sqrt{7}) / 2$ .