Algebra Of Complex Numbers

We are tasked with simplifying the Boolean expression step by step using the laws of Boolean algebra. The given expression is:
Step 1: Group the terms We rewrite the expression for clarity by grouping terms:
Step 2: Apply the Distributive Law to the second group In , factor out :
Using the Complement Law, :
Using the Identity Law, :
Step 3: Substitute back into the main expression Replace with : By the Distributive Law, this is equivalent to:
Step 5: Apply the Absorption Law Using the Absorption Law, . This simplifies the expression to:

Final Simplified Expression:

We have the quadratic equation . First, divide the entire equation by 2 to simplify: Now, factor the quadratic: Setting each factor equal to zero: Thus, the roots are and , corresponding to option (3).