This presentation will attempt to guide you through the information needed to solving harder equations of the type ax 2 + 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 ax 2 + bx + c = 0

There are two types of this kind of equation: 1) Those that can be factored.2)Those that can not. x 2 + 4x - 12 = 0 (x - 2 ) ( x + 6) = 0 x 2 + 4x - 11 = 0 ? Can not factor!! (x - 2 ) = 0( x + 6) = 0 or 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 ) = 12 Which is the same as …. (x + 2 ) 2 = 16 Taking the square root of both sides…… so….. x = 4 – 2 or x = x = 2 or - 6

Before we go further, lets just check our answers. The questions was: (x + 2 ) = 12 The answer is: x = 2 or – 6. If we substitute the answers back into the equations…. (x + 2 ) = 12 Let’s check -6 We get ( ) = (-4) = 12 You can check x= 2 too!!

We just did the equation (x + 2 ) = 12 The steps were: 1Get ( x + 2) 2 by itself 2take square root of both sides (don’t forget ) 3Solve for x getting two answers.

Lets do the next problem the same way…. (x + 2 ) = 0 Step 1: get the square part alone… (x + 2 ) 2 = 5 Step 2:square root both sides Step 3:solve for 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 ) = 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

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 x + 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 x + 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 x + 25 = ( x + 5) 2 x 2 + bx + b 2 = ( x + b/2) 2

Now lets look back at the question that we had before. x 2 + 4x - 1 = 0 Look at the x partThese 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 = Move 1 to get x part alone MATCHES!! And since x 2 + 4x + 4 = (x + 2) 2 Add 4 to both sides so that it matches what I want

(x + 2) 2 = 5 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!! x x - 12 = 0 We need ( x + b) 2 x x What is “b” ? I hope you said 5. So we want to use ( x + 5) 2 = x x + 25 Which is close to what we want….. x x - 12 = 0 x x = x x + 25 = 37 Look at the x parts

x x + 25 = 37 We now finish by the same way….think what the steps are before clicking!! (x + 5) 2 = 37

If we have an equation that can not be factored, for ex. x x - 5 = 0 Get “x part” alone Set up the square part (x + ) 2 = x 2 –12x Match this up With what we want… So just add 36 to both sides of the equals… +36 If you do not know the trick on how to find the number that goes here then click here.number x x = 5

x x + 36 = (x - 6) 2 = 41

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 a x 2 + b x + c = 0 Can be solved by the equation: 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 x - 5 = 0 ax 2 + bx + c = 0 compare To see that (1) (-12) (-5) a=1b= -12c=-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….

The two places that students often make mistakes are - ( - 12) is positive since b is negative!! - 4 (5) will be positive 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 x 2 ) 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 IV YOU 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 hereclick 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 heart and 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…