When the solutions are going to be complex, which methods can be used to solve quadratic equations?

You could complete the square to state the vertex. 
You could use the quadratic equation to find the roots (which are complex). 

Try an example that will require both.

y = x^2 + 2x + 5

Step One
Get the graph. That's included below.

Step Two
Provide the steps for completing the square.
Note: we should get (-1,4)

y= (x^2 +2x ) + 5
y = (x^2 +2x + 1) + 5 - 1
y = (x +1)^2 + 4

The vertex is at (-1,4)

Step Three
Find the roots. Use the quadratic equation. Note that the graph shows us that the equation never crosses or touches the x axis. The roots are complex.



a = 1
b = 2
c = 5





x = -1 +/- 2i

x1 = -1 + 2i
x2 = -1 - 2i And we are done.