Question

Solve \(x^{2}+2 x+2=0\) in the complex number system.

Short answer

The solutions are \(x = -1 + i\) and \( x = -1 - i\).

Step-by-step solution

Identify the quadratic equation

The given equation is a quadratic equation: In standard form, it is written as: \[ x^2 + 2x + 2 = 0 \]

Calculate the discriminant

The discriminant of a quadratic equation \(ax^2 + bx + c = 0\) is given by \(D = b^2 - 4ac\). For our equation: \(a = 1\), \(b = 2\), and \(c = 2\). Thus, the discriminant is \[ D = 2^2 - 4(1)(2) = 4 - 8 = -4 \]

Interpret the discriminant

Since the discriminant \(D\) is less than 0, the quadratic equation has two complex solutions. These solutions are given by the formula: \[ x = \frac{-b \pm \sqrt{D}}{2a} \]

Apply the quadratic formula

Substitute the values into the quadratic formula: Since \(D = -4\), we have \(\sqrt{D} = \sqrt{-4} = 2i\), where \(i\) is the imaginary unit.Thus, \[ x = \frac{-2 \pm 2i}{2(1)} = \frac{-2 \pm 2i}{2} = -1 \pm i \]

Simplify the solutions

The solutions are simplified to: \[ x = -1 + i \] and \[ x = -1 - i \]

Key concepts

quadratic equations

Quadratic equations are a type of polynomial equation of the form \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are constants, and \(x\) represents the variable. In the given equation \(x^2 + 2x + 2 = 0\), our coefficients are \(a = 1\), \(b = 2\), and \(c = 2\).
To solve quadratic equations, you can use various methods like factoring, completing the square, or the quadratic formula. In complex numbers, the quadratic formula is particularly helpful. The general form of the quadratic formula is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
The key to understanding which method to use often depends on the discriminant.

discriminant

The discriminant helps determine the nature of the solutions for a quadratic equation. It is given by the formula: \[ D = b^2 - 4ac \]
For the equation \(x^2 + 2x + 2 = 0\), we substitute \(a\), \(b\), and \(c\): \(b = 2\), \(a = 1\), and \(c = 2\). This gives us: \(D = 2^2 - 4(1)(2) = 4 - 8 = -4\).
The discriminant can take various values that tell us different things:

• If \(D > 0\), the equation has two distinct real roots.
• If \(D = 0\), there is exactly one real root (also called a repeated root).
• If \(D < 0\), the equation has two complex solutions.

In our problem, \(D = -4\), so we will have complex solutions.

complex solutions

Complex solutions occur when the discriminant \(D\) is less than zero. The quadratic formula is still used, but we need to work with imaginary numbers. To find complex solutions, we use the formula: \[ x = \frac{-b \pm \sqrt{D}}{2a} \]
Substituting our values from the original equation (\(x^2 + 2x + 2 = 0\)), we have: \(a = 1\), \(b = 2\), and \(D = -4\). Hence: \[ x = \frac{-2 \pm \sqrt{-4}}{2} \] Since \(\sqrt{-4} = 2i\) (where \(i\) is the imaginary unit), we get: \[ x = \frac{-2 \pm 2i}{2} = -1 \pm i \] This means our solutions are \(x = -1 + i\) and \(x = -1 - i\).
These are complex solutions because they include the imaginary unit \(i\).

imaginary numbers

Imaginary numbers are numbers that, when squared, give a negative result. The most basic imaginary number is \(i\), defined as \(i^2 = -1\). This means:

• \(\sqrt{-1} = i\)
• \(\sqrt{-4} = 2i\)

Using imaginary numbers we can solve equations that do not have real solutions. In our case, since the discriminant is negative (\(D = -4\)), we use \(i\) to find the complex solutions: \(-1 + i\) and \(-1 - i\).
This helps us understand that even if a quadratic equation doesn’t have real solutions, it can still have complex ones. These solutions are just as valid and provide a deeper understanding of mathematics.
Imaginary numbers combined with real numbers are called complex numbers, which take the form \(a + bi\), where \(a\) and \(b\) are real numbers.