SOLUTION: What three techniques can be used to solve quadratic equations? Demonstrate these techniques on the equation 12x^2-10x-42=0.

Question 149608: What three techniques can be used to solve quadratic equations? Demonstrate these techniques on the equation 12x^2-10x-42=0.
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Technique #1 Factoring:

First let's factor

Start with the given expression

Factor out the GCF

Now let's focus on 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 :
1,2,3,6,7,9,14,18,21,42,63,126
-1,-2,-3,-6,-7,-9,-14,-18,-21,-42,-63,-126

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

These factors pair up and multiply to .
1*(-126)
2*(-63)
3*(-42)
6*(-21)
7*(-18)
9*(-14)
(-1)*(126)
(-2)*(63)
(-3)*(42)
(-6)*(21)
(-7)*(18)
(-9)*(14)

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	-126	1+(-126)=-125
2	-63	2+(-63)=-61
3	-42	3+(-42)=-39
6	-21	6+(-21)=-15
7	-18	7+(-18)=-11
9	-14	9+(-14)=-5
-1	126	-1+126=125
-2	63	-2+63=61
-3	42	-3+42=39
-6	21	-6+21=15
-7	18	-7+18=11
-9	14	-9+14=5

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 factors to

Set the factored expression equal to zero

Now set each factor equal to zero:

or

or Now solve for x in each case

So our answers are

or

Technique #2 Quadratic Formula:

Start with the given equation.

Let's use the quadratic formula to solve for x

Start with the quadratic formula

Plug in , , and

Negate to get .

Square to get .

Multiply to get

Rewrite as

Add to to get

Multiply and to get .

Take the square root of to get .

or Break up the expression.

or Combine like terms.

or Simplify.

So our answers are or

Technique # 3 Completing the square

Start with the given expression

Factor out the leading coefficient

Take half of the x coefficient to get (ie ).

Now square to get (ie )

Now add and subtract this value inside the parenthesis. Notice how . Since we're adding 0, we're not changing the equation.

Now factor to get

Combine like terms

Distribute

Multiply

So after completing the square, becomes .

So is equivalent to

Start with completed square equation.

Add to both sides.

Divide both sides by 12.

Take the square root of both sides.

or Break up the expression

or Take the square root of to get

or Subtract from both sides.

or Combine like terms and simplify.

So the answers are or

Technique # 4 Graphing

Simply graph to get

Graph of

Now use the calculator's zero function to find the zeros at and
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