Question

Solve each equation in the complex number system. $$ x^{2}+4 x+8=0 $$

Short answer

The solutions are \( x = -2 + 2i \) and \( x = -2 - 2i \).

Step-by-step solution

Write down the quadratic equation

The given equation is \[ x^{2} + 4x + 8 = 0 \]

Identify coefficients

Identify coefficients from the equation \( ax^{2} + bx + c = 0 \). Here, \( a = 1 \), \( b = 4 \), and \( c = 8 \).

Apply the quadratic formula

The quadratic formula is \[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \].

Calculate the discriminant

Calculate the discriminant \( \Delta = b^{2} - 4ac \). Substituting the values, we get: \[ \Delta = 4^{2} - 4(1)(8) = 16 - 32 = -16 \].

Substitute in the quadratic formula

Since the discriminant is negative, the solutions will involve complex numbers. Substitute the values into the formula: \[ x = \frac{-4 \pm \sqrt{-16}}{2(1)} = \frac{-4 \pm 4i}{2} \].

Simplify the solutions

Simplify to obtain the solutions: \[ x = \frac{-4 + 4i}{2} = -2 + 2i \] and \[ x = \frac{-4 - 4i}{2} = -2 - 2i \].

Key concepts

Complex Numbers

In mathematics, complex numbers are numbers that have both a real part and an imaginary part. The standard form of a complex number is written as ( a + bi ), where ( a ) is the real part, ( b ) is the imaginary part, and ( i ) is the imaginary unit. The imaginary unit ( i ) is defined by the property ( i^2 = -1 ). Complex numbers are particularly useful when solving quadratic equations that do not have real solutions.
In our problem, the discriminant is negative, indicating the presence of complex solutions. This is why we end up with ( x = -2 + 2i ) and ( x = -2 - 2i ). Recognizing this step is crucial in solving equations like ( x^{2} + 4x + 8 = 0 ) effectively.

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations of the form ( ax^2 + bx + c = 0 ). The formula is expressed as:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula gives solutions by plugging in the values of the coefficients ( a, b, \text{ and } c ) from the quadratic equation.
By following the quadratic formula, we can find all solutions, including those that involve complex numbers. The plus-minus ( ± ) symbol indicates that there are generally two solutions. For our example, with ( a = 1 ), ( b = 4 ), and ( c = 8 ), using the formula gave us the complex solutions.

Discriminant Calculation

The discriminant, denoted as ( \triangle ), plays a crucial role in determining the nature of the roots of a quadratic equation. The discriminant is calculated by the expression:
\[ \triangle = b^2 - 4ac \]The value of the discriminant indicates the type of roots:

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

For the given equation, we calculated ( \triangle ) as:\[ 4^2 - 4(1)(8) = 16 - 32 = -16 \]This negative discriminant tells us that the quadratic equation has complex roots.

Coefficients Identification

Identifying the coefficients in a quadratic equation is an essential first step in solving the equation using the quadratic formula. A quadratic equation is generally written in the form ( ax^2 + bx + c = 0 ), where:

• ( a ) is the coefficient of ( x^2 ).
• ( b ) is the coefficient of ( x ).
• ( c ) is the constant term.

In our example, the equation provided is:
\[ x^2 + 4x + 8 = 0 \]By comparing it to the standard form, we can identify the coefficients as:
a = 1
b = 4
c = 8Once you have correctly identified the coefficients, you can substitute them into the quadratic formula to find the solutions of the equation.