SOLUTION: Quadratic equations may be solved by graphing, using the quadratic formula, completing the square, and factoring. What are the pros and cons of each of these methods? When might ea

Question 279999: Quadratic equations may be solved by graphing, using the quadratic formula, completing the square, and factoring. What are the pros and cons of each of these methods? When might each method be most appropriate? Which method do you prefer? Explain why.

Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
Graphing method.
The graphing method is most useful as a visual learning tool. With the graphing method you can see what the solutions represent. But as a means of finding the solution, there is only one circumstance under which I would consider using this method: I had a graphing calculator with a tracing facility and decimal approximations of the solutions were satisfactory. Otherwise I don't even think the graphing method is good for solving linear equations (and quadratics are harder to graph!) Here's what I don't like about this method:

It's accuracy depends entirely on...
- the accuracy of the graph
- the resolution of your vision
Even with a perfect graph and marvelous vision, you will
- never find irrational solutions.
- never find complex solutions.
- have great difficulty identifying solutions which are not integers.
Very large solutions (like 32904 and -2000000) will require impossibly huge graphs.

Completing the square method.
Completing the square works perfectly well for solving quadratic equations and it is very useful in other areas of Math like conic sections (parabolas, circles, ellipses, hyperbolas) and Calculus. Because of all this it is good to learn how to complete the square. However, I rarely suggest or recommend it use for solving quadratic equations because:

Compared to factoring and the Quadratic Formula, completing the square:
- is not easy.
- is a lot of work
- is prone to errors, especially at the point where you find the square root of each side (because you forget that there is both a positive and negative square root).
The Quadratic Formula is the result of completing the square of the general quadratic equation, , so why not save the effort and avoid the possible errors and use the formula instead?

Factoring Method.

Pluses:
- Fast (usually)
- Finds all rational solutions.
- Works on higher degree equations, too.
- Gives you good practice at a skill, factoring, that is very important in many areas in Math, including:
  - Solving quadratic and higher degree equations
  - Reducing fractions
  - Finding Lowest Common Denominators and Least Common Multiples
Minuses:
- Factoring can be difficult sometimes.
- Irrational and complex solutions are impossible (or nearly impossible) to find with this method.

Quadratic formula.

Pluses:
- It finds solutions of any kind: rational, irrational or complex.
Minuses:
- You have to memorize the formula
- You have to know where to find the "a", the "b" and the "c" in the equation.
- The "plus or minus" can be confusing.
- The square root can be difficult for some.
- The fraction can cause problems for some.
- It's a lot of work for rational solutions (compared to the factoring method).

So when I try to solve a quadratic equation I first try to use factoring. If the equation doesn't factor (or is extremely difficult to factor) then I use the Quadratic Formula. (In very rare cases, if an equation is already in a form where completing the square is or is almost already finished, then I would consider using completing the square. (For example: .)) I never even think of using the graphing method.
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