SOLUTION: What are the different methods for solving quadratic equations? Give an example. Please help me and thank you!

Question 897456: What are the different methods for solving quadratic equations? Give an example. Please help me and thank you!
Answer by richwmiller(17219) (Show Source): You can put this solution on YOUR website!
Factor [Factorize]
Complete the square
Use quadratic formula

Factor by grouping

Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

Start with the given expression.

Factor out the GCF .

Now let's try to factor the inner expression

---------------------------------------------------------------

Looking at the expression , we can see that the first coefficient is , the second coefficient is , and the last term is .

Now multiply the first coefficient by the last term to get .

Now the question is: what two whole numbers multiply to (the previous product) and add to the second coefficient ?

To find these two numbers, we need to list all of the factors of (the previous product).

Factors of :

1,3

-1,-3

Note: list the negative of each factor. This will allow us to find all possible combinations.

These factors pair up and multiply to .

1*3 = 3
(-1)*(-3) = 3

Now let's add up each pair of factors to see if one pair adds to the middle coefficient :

First Number	Second Number	Sum
1	3	1+3=4
-1	-3	-1+(-3)=-4

From the table, we can see that the two numbers and add to (the middle coefficient).

So the two numbers and both multiply to and add to

Now replace the middle term with . Remember, and add to . So this shows us that .

Replace the second term with .

Group the terms into two pairs.

Factor out the GCF from the first group.

Factor out from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.

Combine like terms. Or factor out the common term

--------------------------------------------------

So then factors further to

===============================================================

Answer:

So completely factors to .

In other words, .

Note: you can check the answer by expanding to get or by graphing the original expression and the answer (the two graphs should be identical).

Complete the square

Solved by pluggable solver: COMPLETING THE SQUARE solver for quadratics

Read this lesson on completing the square by prince_abubu, if you do not know how to complete the square.
Let's convert to standard form by dividing both sides by 1:
We have: . What we want to do now is to change this equation to a complete square . How can we find out values of somenumber and othernumber that would make it work?
Look at : . Since the coefficient in our equation that goes in front of x is -1, we know that -1=2*somenumber, or . So, we know that our equation can be rewritten as , and we do not yet know the other number.
We are almost there. Finding the other number is simply a matter of not making too many mistakes. We need to find 'other number' such that is equivalent to our original equation .

The highlighted red part must be equal to -2 (highlighted green part).

, or .
So, the equation converts to , or .

Our equation converted to a square , equated to a number (2.25).

Since the right part 2.25 is greater than zero, there are two solutions:

, or

Answer: x=2, -1.

Use quadratic formula

Solved by pluggable solver: Quadratic Formula

Let's use the quadratic formula to solve for x:

Starting with the general quadratic

the general solution using the quadratic equation is:

So lets solve ( notice , , and )

Plug in a=1, b=2, and c=3

Square 2 to get 4

Multiply to get

Combine like terms in the radicand (everything under the square root)

Simplify the square root (note: If you need help with simplifying the square root, check out this solver)

Multiply 2 and 1 to get 2

After simplifying, the quadratic has roots of

or

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