Questions: Solve the equation in the complex number system. x^2-49=0 What is the solution set? x= (Use a comma to separate answers as needed.)

Solve the equation in the complex number system. x^2-49=0 What is the solution set? x= (Use a comma to separate answers as needed.)
Transcript text: Solve the equation in the complex number system. \[ x^{2}-49=0 \] What is the solution set? $x=\{$ $\square$ (Use a comma to separate answers as needed.)
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Solution

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Solution Steps

To solve the equation \(x^2 - 49 = 0\) in the complex number system, we can use the difference of squares formula. The equation can be rewritten as \((x - 7)(x + 7) = 0\). This implies that \(x - 7 = 0\) or \(x + 7 = 0\). Solving these equations will give us the solutions for \(x\).

Step 1: Identify the Equation

We start with the equation \(x^2 - 49 = 0\).

Step 2: Apply the Difference of Squares

The equation can be rewritten using the difference of squares formula: \[ x^2 - 49 = (x - 7)(x + 7) = 0 \]

Step 3: Solve for \(x\)

Set each factor equal to zero and solve for \(x\):

  1. \(x - 7 = 0 \Rightarrow x = 7\)
  2. \(x + 7 = 0 \Rightarrow x = -7\)

Final Answer

The solution set is \(\boxed{x = \{-7, 7\}}\).

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