This presentation will attempt to guide you through the information needed to solving harder equations of the type

ax2 + bx + c = 0

When you get to the part that says Worksheet, you will get your worksheet out and attempt the problems. Only do the part that you are directed to do. The answers are at the front of the room. Check them yourself.

If you get them right, good for you!!! You may continue to the next part.

If you make mistakes and do not understand you can ask me or a group member. But you must understand before you go on to the next section.

Solving equations of the type

ax2 + bx + c = 0

There are two types of this kind of equation:

1) Those that can be factored. 2)Those that can not.

x2 + 4x - 12 = 0

(x - 2 ) ( x + 6) = 0

x2 + 4x - 11 = 0

? Can not factor!!(x - 2 ) = 0 ( x + 6) = 0or

So x = 2 or -6

Since the type that can be factored is easy, lets find a way to

do the other type. Lets start with ……..

(x + 2 ) 2 - 4 = 12

Which is the same as ….

(x + 2 ) 2 = 16

Taking the square root of both sides……

162 2 x 42 x

so…..

x = 4 – 2 or x = - 4 - 2

x = 2 or - 6

Before we go further, lets just check our answers.

The questions was:

(x + 2 ) 2 - 4 = 12

The answer is: x = 2 or – 6.

If we substitute the answers back into the equations….

(x + 2 ) 2 - 4 = 12

Let’s check -6

We get (-6 + 2 ) 2 - 4 =

(-4) 2 - 4 = 12

You can check x= 2 too!!

We just did the equation

(x + 2 ) 2 - 4 = 12

The steps were:1 Get ( x + 2) 2 by itself

2 take square root of both

sides (don’t forget )

3 Solve for x getting two answers.

Lets do the next problem the same way…. (x + 2 ) 2 - 5 = 0

Step 1: get the square part alone…

(x + 2 ) 2 = 5

Step 2: square root both sides

52 2 x

25 xStep 3: solve for x

2525 xorx

52 x

Since you understand get your worksheet and do only part I. Check your own answers.

If you did not understand what we have done so far, go back and look at the previous slides again or ask MR. Erickson BUT DO NOT GO FURTHER

If you did understand then continue

And since they are the same equation, they have the same answer!

Welcome back!!! Let’s look at the problem we just did.

(x + 2 ) 2 - 5 = 0

But this is the same as

Is the same as

(x+2)(x+2) - 5 = 0

(x 2 + 2x + 2x + 4) - 5 = 0

x 2 + 4x - 1 = 0

So if I give you the equation

x 2 + 4x - 1 = 0

You should be able to give me the answer

2525 xorx

The trick is being able to get the squared part!

To give you an idea on how to get this lets look at some squares and see if we can find any patterns….

( x + 2) 2 = x 2 + 4x + 4 ( x + 3) 2 = x 2 + 6x + 9 ( x + 5) 2 = x 2 + 10x + 25

Look at the blue numbers and compare them with the red one. What do you notice? Does it work for negatives as well?

( x - 7) 2 = x 2 - 14x + 49

Excellent. You have done well. Now, if you think you have the stomach continue down the path but be careful that you don’t get too confident!! Many a brave person has perished down this same journey.

So, if you think you understand the pattern fill in the answers to questions

part II on your worksheet RIGHT NOW. Check your answers. If you have it wrong then ask a group member or ask me!!

If you got the questions correct then a big “CHEERS” to you!! Well done!!!. But danger still lurks around the corner. For those brave of heart, continue on…….

Good job. You have made it to the next step. Are you sure you have what it takes? We shall see!

We should be able to go both ways too….

x 2 + 4x + 4 = ( x + 2) 2

x 2 + 10x + 25 = ( x + 5) 2

x 2 + bx + b2 = ( x + b/2) 2

Now lets look back at the question that we had before.

x 2 + 4x - 1 = 0

Look at the x part These are the important parts: x 2 + 4x

I want my equation to look something like ( x + b) 2 = 7. Then I can finish it like I did on the worksheet by taking the square root!!

So, what number should I choose to put in “b” to make sure that we will get “4x” ?

I hope that you chose 2!!

Why?Because……..

I want to change this “x part” to look like (x + b ) 2

The problems is x 2 + 4x - 1 = 0

(x+2) 2

Let’s see why 2 is nice

(x + 2) 2 = x 2 + 4x + 4

Almost matches!! But, it is very close!! we get the x parts

x 2 + 4x

Since the x parts match, it is now easy to make the number part match

too!! Watch!!

x 2 + 4x - 1 = 0

(x + 2) 2 = 5

And now we finish like before…..

x 2 + 4x = 1

x 2 + 4x + 4 = 1 + 4

Move 1 to get x part alone

MATCH

ES!!And since x 2 + 4x + 4 = (x + 2) 2

Add 4 to both sides so that it matches what I want

52 2 x

(x + 2) 2 = 5

2525 xorx

But you still might want to back and look at the steps again to remind yourself…

Another one….. But this time you try to think of the next step before clicking!!

x2 + 10x - 12 = 0

We need ( x + b) 2

x2 + 10x

What is “b” ?

I hope you said 5.

So we want to use

( x + 5) 2 = x2 + 10x + 25

Which is close to what we want…..

x2 + 10x - 12 = 0

x2 + 10x = 12

+25 +25

x2 + 10x + 25 = 37

Look at the x parts

x2 + 10x + 25 = 37

We now finish by the same way….think what the steps are before clicking!!

(x + 5) 2 = 37

375 2 x

537537 xorx

If we have an equation that can not be factored, for ex.

x 2 - 12x - 5 = 0

Get “x part” aloneSet up the square part

(x + ) 2 = x 2 –12x

Match this upWith what we want…

So just add 36 to both sides of the equals…

+36 +36

If you do not know the trick on how to find the number that goes here then click here.

- 6 +36

x 2 - 12x = 5

x 2 - 12x + 36 = 5 + 36

(x - 6) 2 = 41

416 2 x

641641 xorx

This method that you just learned is called “completing the square”

Go now to your worksheet and do

part III (or you may go back and review the previous slides if you need more help)

Congratulations that you made it this far!!!

But far from over it is!! Take care!!!

Now, if you were wondering if you have to do this long process every time on these kind of problems, your answer is ………..

There is a formula!!! Are you happy?

Welcome back those with true and courageous hearts!!!

Solving equations of the type ax2 + bx + c = 0

Can be solved by the equation:

a

acbbx

2

42

Where you can find the a, b and c from the problem!!! (“0” must be on one side of the equals!!!)

Ex (the one we just did)x 2 - 12x - 5 = 0

ax2 + bx + c = 0compareTo see that

a

cabbx

2

42

(1)

(1)

(-12) (-12) (-5)

a=1 b= -12 c=-5

**Did you notice that in this example both “b” and “c” are negative? Can you see why? Also, a = 1 and NOT 0!! This is a

common place for mistakes with students….

)1(2

)1)(5(4)12()12( 2 x

2

16412

2

2014412

x

The two places that students often make mistakes are

- ( - 12) is positive since b is negative!!

- 4 (5) will be positive

2

41212x

641641 xorx

Let’s finish with simple calculations

1 get 0 on one side of the equation to use the formula (like factoring!)

2 Find a (next to x2) b (next to x) and c the lonely guy

3 Substitute into formula- careful if b and/or c are negative because you will get double negatives

4 You should have 2 answers when solving quadratics (the two answers may be the same!)

5 If the number under the square root (called the “determinate”) is negative, there is no solution.

Things to remember, says George

You have shown your worth, that is definite!!. I have to say that you are close to the end and you have not yet been deterred by all the evil set forth!!!

BUT IT IS NOT OVER YET!!!!

Your turn. Get out the worksheet

and try part IVYOU MAY WANT

TO PRAY!!

You should know three ways

to solve quadratic

equations now.

Factoring (but this does not always work

Quadratic formula (always works and easy)

What should you have learned?

Completing the square (difficult)

If you want to review this part click here

A QUICK REVIEW

Now you have a few more “thinking”questions left on your worksheet. Do not proceed until you have finished those questions…..

You have done well to get this far. It is a lot of information so if you understood it all you should be proud!!!

Once you finish the whole worksheet you can………………

And this is where many gallant have fallen never to return. Remember, only the true of heartand pure of mind will survive. Call to Allah for help while you can!!

All you have to do is divide the coefficient of the “x” term by 2.

So, in this example…….… -12/2 = -6

Keep click to back to go back to the slide you were before…