Algebra 1 The Quadratic Formula
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Magic Formula
The Quadratic Formula

 
 So now we have a step by step process to find X,
 but wow! it is really long and involved.
 It is a lot to remember!
 Couldn't we maybe just have a formula
 to solve these things all in one shot?
 Huh?
 Please?
 
 SURE
 
 For the last two pages, we've been solving stuff like:
 

 3X2 + 5X + 2 = 0

 and

 X2 - 4X + 3 = 0

 

 Here's what we're going to do.
 We're going to write one of these equations as ...
 

AX2 + BX + C = 0

 
 Then we're going to do our long process from the last page on it.
 In the end, we will have ...
 

 X = some junk with A's B's and C's in it

 
 From then on, when we have one of these equations
 all we have to do is pick out the A, B, and C value
 and put it in the formula we are about to build.
 
 No more nasty factoring.

 Just remember the formula, and you're set.

 Here we go: 
 
 STEP 1:
 Divide both sides of the equation by the coefficient on the X2 ...
 

 
 Simplify ...
 

 
 STEP 2:
 Take the X term and determine the value of B.
 Use that to find B2.
 

 
 STEP 3:
 Put the B 2 value into the equation.
 

 
 STEP 4:
 Gather up the terms of the square, 
 and write them as the square.
 

 
 STEP 5:
 Put all of the other baloney that's not part of the square 
 on the right side of the equation.
 Simplify it as much as you can.
 

 
 STEP 6:
 Solve this for X.
 

 
 And there it is!
 
 The most important formula in all of Algebra.
 
 It was a lot of work getting here,
 but here is the payoff.
 
 Now, if we have something like ...
 

5X2 - 13X + 6 = 0

 
 We can say, OK:
 

A = 5          B = -13          C = 6

 
 and put these numbers into "THE FORMULA"
 

 
 This is the same answer as we got before for this one,
 but this time it was a whole lot easier.
 
 Our new formula is called ...
 

The Quadratic Formula

 
 When we use it, sometimes we will get two answers,
 sometimes we will get one answer,
 and sometimes we will not get any answers.
 
 If we don't get any answers, 
 it doesn't mean that the formula didn't work.
 It means that the problem really doesn't have any answers.
 
 There is a quick way to tell how many answers you will get.
 Just look at this part of the formula:
 

 
 If B 2 - 4AC works out to be a positive number,
 you will get two answers for X.
 
 If B 2 - 4AC works out to be zero,
 you will get one answer for X.
 
 If B 2 - 4AC works out to be a negative number,
 you will not get any answers for X.
 
 Because this piece of the formula can tell you that,
 it gets it's own name.
 It is called:
 

The Discriminant

 
 Let's do another one ...
 
 Example:
 

4X2 - 20X + 25 = 0

 
 So:

A = 4          B = -20          C = 25

 

 
 The "discriminant" part of the formula was equal to zero,
 so we only get one answer.
 
 Example:
 

6X2 = - 2X + 4

 
 Whoa! Time out!
 Before we do anything, we need to move all of the values 
 over to the left side of the equation so we have:
 

AX2 + BX + C = 0

 

 
 So:
 

A = 6          B = 2          C = -4

 

 

   copyright 2005 Bruce Kirkpatrick

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