Question

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

Short answer

x = 2, -2

Step-by-step solution

Simplify the equation

Rewrite the equation so it is set equal to 0: \[ x^2 - 4 = 0 \]

Recognize the factoring form

Notice that the left-hand side of the equation can be factored using the difference of squares: \[ x^2 - 4 = (x - 2)(x + 2) \]

Set each factor to zero

For the equation to be true, each factor must equal zero: \[ x - 2 = 0 \] and \[ x + 2 = 0 \]

Solve each equation

Solving each of the equations for x gives: \[ x - 2 = 0 \rightarrow x = 2 \] \[ x + 2 = 0 \rightarrow x = -2 \]

List the solutions

The solutions to the equation \[ x^2 - 4 = 0 \] are \[ x = 2 \] and \[ x = -2 \].

Key concepts

Complex Number System

The complex number system extends our understanding of numbers beyond the real line. A complex number is expressed as \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit with the property \(i^2 = -1\). This system helps address solutions that don't exist within the real numbers alone. For quadratic equations with no real solutions, complex numbers come into play. However, in the given exercise \(x^2 - 4 = 0\), all solutions are real. If the equation were \(x^2 + 4 = 0\), the solutions would be complex: \(x = \pm 2i\).

Difference of Squares

The difference of squares is a special algebraic form: \(a^2 - b^2 = (a - b)(a + b)\). This pattern is crucial when solving quadratic equations because it allows for easy factoring. In the given problem, \(x^2 - 4\) can be written as \((x)^2 - (2)^2\). Hence, it factors to \( (x - 2)(x + 2) \). Recognizing this pattern simplifies solving the equation significantly.

Factoring Quadratic Equations

Quadratic equations that can be factored into simpler binomial expressions are easier to solve. The basic form of a quadratic equation is \(ax^2 + bx + c = 0\). For \(x^2 - 4 = 0\), the equation directly factors using the difference of squares. Here's how to factor such equations:

• Determine if the equation fits the form \(a^2 - b^2\).
• Rewrite the expression as \((a - b)(a + b)\).
• Set each factor equal to zero and solve for \(x\).

Factoring transforms the problem into simpler linear equations, making it straightforward to find the solutions.

Solving Linear Equations

Solving linear equations involves finding the value of the variable that makes the equation true. After factoring the quadratic equation \(x^2 - 4 = 0\) to \( (x - 2)(x + 2) = 0 \), set each factor to zero:
\(x - 2 = 0\)
\(x + 2 = 0\)

• Solve each equation separately.
• For \(x - 2 = 0\), add 2 to both sides to get \(x = 2\).
• For \(x + 2 = 0\), subtract 2 from both sides to get \(x = -2\).

This gives the final solutions: \(x = 2\) and \(x = -2\).