In the earlier two sections we've talked quite a bit regarding solving quadratic equations. A logical question to ask at this point is which method has to we employ to solve a given quadratic equation? Unluckily, the answer is, it depends.
If your instructor has mentioned the method to employ then that, of course, is the method you have to use. Though, if your instructor had not indicated the method to apply then we ought to make the decision ourselves. Following is a general set of guidelines which may be helpful in determining which method to employ.
1. Is it obviously a square root property problem? In other terms, does the equation consist only of something squared and a constant? If it is true then the square root property is probably the simplest method for use.
2. Does it factor? If so, this is possibly the way to go. Notice that you must not spend a lot of time attempting to determine if the quadratic equation factors. Look at the equation & if you can rapidly determine that it factors then go with that. If you can't rapidly determine that it factors then don't worry regarding it.
3. If you've attained this point then you've determined that the equation is not in the right form for the square root property & that it don't factor (or that you can't rapidly see that it factors). Thus, at this point you're only real choice is the quadratic formula.
Once you've solve sufficient quadratic equations the above set of guidelines will become almost second nature to you & you will determine yourself going through them almost without thinking.
Notice that nowhere in the set of guidelines was carrying out the square mentioned. The cause for this is easily that it's a long method which is prone to mistakes while you get in a hurry. The quadratic formula will always also work and is much shorter of method to employ. Generally, you must only use completing the square if your instructor has needed you to use it.
As a solving technique completing the square must always be your last choice. It doesn't mean though that it isn't significant method. This process is very useful in many situations of which solving is only one.
In the previous we saw that solving a quadratic equation in standard form,
ax^{2} + bx + c =0
We will obtain one of the following three possible solution sets.
1. Two real distinct (i.e. not equal) solutions.
2. A double root. Remember this arises while we can factor the equation into a perfect square.