## Introduction

Hello and welcome to my smore. In this, i will be helping you in factoring, expanding, completing the square, and solving. I hope you benefit from this and get a good mark on your test. Enjoy!!

## Expanding

Expanding is a form of getting 2 numbers and variable multiplying them (using distributive property) to end up with a standard form quadratic. Expanding is just taking the factored form and putting it back to its original form. Here is an example:

I have a variable and a number in brackets multiplied by another variable and a number in brackets.

(x-12) (x+3)

Now I am going to use distributive property to put it in standard form.

x times x is X2

x times 3 is 3x

-12 times x is -12x

-12 times 3 is -36

If you put it all together, you will get X2 - 9x -36

If you are still stuck, look at the image below. It is another example.

## Factoring

Factoring is basically finding what to multiply to get an expression. In a simple quadratic formula, you have to find 2 numbers that multiply to the c value and what adds to the b value. For example:

Simple Trinomial

I have a equation like this x2 + 5x + 6

So the question is what 2 numbers multiply's to 6 and add to 5. The answer is 3 and 2. This is because 2 times 3 is 6 and 2 plus 3 is 5. Easy right?

Now we have to put it in the expanded form which is like this:

(x+3)(x+2) -- if you expand this you will get the same equation as the one above. It is a cycle!

Decomposition

Now this is the simplest form. There are still 2 more forms to cover. The next one is called decomposition.

If I have the same equation as the one above but instead of nothing in front of the A value, I have a number. This is then when we use decomposition.

So lets say I have this :

2x2 + 5x + 3 - I cant factor this because there is nothing in common. Therefore, I will multiply the A value by the C value and do the same thing as before; which is to find 2 numbers that multiply to the (new value) and add to the B value.

ok so 2 times 3 is 6. What multiply's to 6 and adds to 5. 3 and 2. Now we put it like this:

2x2 +3x +2x +3 - what you do is that you remove the b value and put the 2 new numbers in and add an x beside it.

Now we have to factor by grouping. This is a sub category. Basically you take the first 2 numbers/variables and you factor and then you take the next two varialbes/numbers and you factor.

3/2x(4/3x+2)+3/2(4/3x+2)

then if you join the things that are not in brackets together and take out 1 of the same things in brackets.

(3/2x+2)(4/3x+2)

And there you go. By the way sorry, I had to use fractions. Usually fractions/decimals are not used in decomposition.

Difference Of Squares

The last factoring thing is called difference of squares.

Ok so lets say I have an equation like this

x2 - 49y2

If all these can be square rooted and there is a negative sign we can find the difference of squares.

This is what we do.

( x+7y) (x-7y)

We get the root of it and then we do x + whatever the number and x - whatever the number. Also it can have more than 1 variable, different variable, etc. (not necessarily x.)

Still having trouble with difference of squares? Watch This !!

## Completing The Square

Completing the square is simply converting from standard form to vertex form. It is a very simple process and gives you useful information such as the stretch/compress and the vertex. Do do this there are a few steps. Lets use the equation 4x2+ 2x+6 as a example.

1. If there is a A value factor it out. In this case their is

4(x2+0.5x + 1.5)

2. Take the C value out but multiply it by the factored A

4(x2+0.5x)+6

3. Divide the B value by 2 and square it. Then add the answer in the equation.

4(x2+0.5x + 0.0625)+6

4. The same number that you divided by 2 and squared, multiply it by the A. Then place it outside the bracket

4(x2+0.5x + 0.0625)+6 + 0.25 ( Add 6 - .25)

= 4(x2+0.5x + 0.0625)+5.75

5. Now see that value that was divided and squared and added in the brackets. Get rid of it.

= 4(x2+0.5x)+5.75

6. And the last step is to divide the 0.5 or B value by 2 and square the whole bracket. Also get rid of the square on the x

4(x+0.25)^2+5.75

And there you go. You have completed the square successfully!

Still stuck, this video should help.

❤² How to Solve By Completing the Square (mathbff)