Question 136135: Quadratic equations can be solved by graphing, using the guadratic formula, completing the square and factoring. What are the pros and cons of each of these methods? When might each method be most appropriate? Why method do you prefer and why?
Found 2 solutions by solver91311, rizki_kiki:
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website! Graphing has the advantage that you can actually "see" the answer, provided that you have created the graph accurately. The disadvantage is that unless the roots of the equations are nice, neat integers and the graph intersects the x-axis exactly at a tick mark on the axis, you only have an approximation of the value of the roots.
Factoring is a fairly simple process, provided the product and sum of the constant term factors neatly provide you with the information required to reverse the FOIL process. The factoring method is further complicated when the lead coefficient is other than 1 and is not a factor of the coefficients of the other two terms.
Completing the square always works regardless of the nature of the roots and/or the factorability of the quadratic over the rationals or even the reals. However, the complexity and potential for error increases when either the lead coefficient other than 1 and is not a factor of the other coeffients, or the 1st degree term is odd, or both.
The quadratic formula ALWAYS works, and the complexity/error potential is constant regardless of the nature and relationship of the coefficients. In fact, the quadratic formula is the general application of the completing the square process (the quadratic formula was derived by completing the square on the general quadratic:  )
My own personal method is to look quickly at the quadratic. If the factors are obvious, I use that method. If not, I look at the relationship of the coefficients to see if completing the square will be a nice tidy computation. Nice, tidy computations involving integers or fractions composed of small numbers, particularly in the denominator, are always less subject to error than otherwise. If completing the square looks like it will be messy, I use the quadratic formula.
Always calculate the discriminant first when using the quadratic formula. There are times when the factorization is not obvious, or for some reason I just don't see it. The fact that the quadratic is factorable over the rationals becomes obvious if the discriminant turns out to be a perfect square.
I hope that helps. I didn't give you any hard and fast "rules" for chosing a solution method, but that is because there aren't any. Personal preference enters into the equation, and that is very hard to pin down in a set of rules.
Answer by rizki_kiki(1)  (Show Source):
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