Up to this point, we have been working exclusively with real numbers. Now we shed this limitation and allow for a much broader range of problems.

With these definitions, we have greatly expanded our space of numbers. Notice that any real number is also a complex number, for example 5 = 5 + 0

*i*. Here the real part is 5 and the imaginary part is 0. Next we consider powers of

*i*.

**Simplify**.

Adding and subtracting complex numbers is just a matter of adding like terms. Be sure to use the order of operations and add real and imaginary parts separately.

**Add or subtract**.

We have to use a bit of caution when multiplying complex numbers. First, we will run into

*i*^2 often. In this case, we will replace them all with -1. In addition, the property

is true only when either

*A* or

*B* is non-negative; so it is best to simplify using the imaginary unit before multiplying.

**Tip**: Make use of the imaginary unit if the radicand is negative before trying to simplify.

**Multiply**.

**Multiplying complex numbers often requires the distributive property.**
Dividing complex numbers requires techniques similar to rationalizing the denominator.

**Divide**.

When a complex number is in the denominator, multiply numerator and denominator by its conjugate.

Sometimes quadratic equations have complex solutions.

**Solve by extracting roots**.

**Solve by completing the square**.

**Solve using the quadratic formula**.

If when solving for the

*x*-intercepts, the solutions come out complex then there are none. This tells us that the parabola is completely above or below the

*x*-axis.

**Graph and label all important points**:

**Graph and label all important points**:

**Video Examples on YouTube**: