Can you use the quadratic formula for all situations where you would factor or complete the square

Can you use the quadratic formula for all situations where you would factor or complete the square?

Because sometimes I get a negative number under the square symbol and my calculator just gives me an 'Error' message when I try to solve. What do I do in that case?

Thanks

To answer your question completely, yes, you are able to use the Quadratic Equation each and every time. However, it is not always the most convenient.

A negative number under the radical, means you will have 2 imaginary/complex solutions instead of real solutions. They are still in fact solutions, however an average calculator will return an error. If you aren't in Algebra 2, you generally disregard the solution and say "No Real Solutions".

You should double check with your instructor on how he/she wants you to put the answer. Most Algebra 2 courses, or Precal courses cover "Complex Roots" and so they actually arrive at solutions with "i" in them, which represents Sqrt(-1).

Hope that helps!

DErivation of quadratic formula
ax² + bx + c = 0 ----- divide by a
x² + b/a x + c/a = 0 -- (make x² + b/a x ) complete square by adding and subtracting (b²/4a²)
x² + b/a x + b²/4a² – b²/4a² + c/a = 0
x² + b/a x + b²/4a² = b²/4a² – c/a
(x + b/2a)² = b²/4a² – c/a
(x + b/2a) = ± √(b²/4a² – c/a)
x = – b/2a ± √(b² –4ac/4a² )
= – b/2a ± √(b² –4ac)/2a
= [– b ± √(b² –4ac)]/2a
------

The quadratic method is derived from the method of
completing the square, so it will always work.
If you get a negative discriminant, your roots are
going to be complex numbers. So whether you are factoring,
or completing the square, you will not find any real solutions.

In the case where you get a negative or an error, there is no answer. Technically, you could use the quadratic formula for everything, but some equations take less time to factor other ways, and some don't have answers.