What are the solutions of this quadratic equation?
$x^2+2 x=-2$
The problem asks for the solutions to the quadratic equation \(x^2 + 2x = -2\). A quadratic equation is typically written in the standard form \(ax^2 + bx + c = 0\). We can transform our equation by adding \(2\) to both sides:
\[x^2 + 2x + 2 = 0.\]
From this standard form, we identify the coefficients as \(a = 1\), \(b = 2\), and \(c = 2\).
To find the values of \(x\) that satisfy the equation, we use the quadratic formula:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.\]
First, we calculate the discriminant, which is the expression under the square root, \(\Delta = b^2 - 4ac\):
\[\Delta = (2)^2 - 4(1)(2) = 4 - 8 = -4.\]
Since the discriminant is negative, the solutions will involve the imaginary unit \(i\), where \(i = \sqrt{-1}\). Thus, \(\sqrt{-4} = \sqrt{4} \cdot \sqrt{-1} = 2i\).
Now, substitute the values into the quadratic formula:
\[x = \frac{-2 \pm 2i}{2(1)}.\]
Simplifying the expression by dividing both terms in the numerator by the denominator:
\[x = \frac{-2}{2} \pm \frac{2i}{2} = -1 \pm i.\]
This indicates that the two solutions are \(x = -1 + i\) and \(x = -1 - i\). Comparing this to the given options, we find that the correct choice is \(x = -1 \pm i\).
Quadratic equations with imaginary roots frequently appear in engineering and physics, particularly in the study of oscillating systems. For example, when analyzing the motion of a damped spring-mass system, the characteristic equation often results in complex roots. These roots describe not just the frequency of the oscillation, but also how quickly the oscillation dies out over time. Understanding these solutions is essential for designing stable mechanical structures and electrical circuits.