## Modeling in Identities

By making use of the feature of area calculation through multiplication, this is a visual way of showing that the result of multiplication is equal to the value of the area.

Multiplication and Area

$$\displaystyle 3.4$$ ; If we times 3 with 4, we will have found the area (the number of squares) within a rectangle, which has a length of 3 units.

Now, by using this feature, let’s form an identity using the area calculation.

🔻🔻Identities are expressions that are equal to one another. 🔻🔻

## Identical of $$\displaystyle (a+b)²$$

The square of anything is the multiplication of any number by itself. Let’s write it down within brackets next to one another.

$$\displaystyle (a+b)²=(a+b).(a+b)$$

This tells us;

If you are going to create a rectangular shape, one side will be $$\displaystyle a+b$$
and the other will also be $$\displaystyle a+b$$ . When we multiply them by one another $$\displaystyle (a+b).(a+b)$$ the area that is calculated is equivalent to our identity.

We have placed $$\displaystyle a+b$$ on either side, now let’s turn this into a rectangular area.

This area is going to give us our identity.

Let’s divide the area into pieces;

$$\displaystyle (a+b).(a+b)=(a+b)²$$

$$\displaystyle (a+b)²=a²+2ab+b²$$

## Identical of $$\displaystyle (a-b)²$$

$$\displaystyle (a-b)²=(a-b).(a-b)$$

I’m going to create a rectangular area and the length of either side will be $$\displaystyle a-b$$.

I need to subtract $$\displaystyle b$$ unit from a unit. I can do this by cutting ‘$$\displaystyle b$$ unit’ length from ‘$$\displaystyle a$$ unit’  length.

Let me take a square that has the side of a unit, in this way I will have created a side of $$\displaystyle a$$

The lengths of the sides are a and the area is $$\displaystyle a²$$.

Let me now cut ‘$$\displaystyle b$$ unit length’ from ‘$$\displaystyle a$$ unit length’ to create an ($$\displaystyle a-b$$) unit length.

Let’s cut this piece and throw it away.

The area that has been cut away:     $$\displaystyle a.b=ab$$

Let’s get rid of the area that has been cut from the complete square. I can reach an identical through ;

$$\displaystyle a²-ab$$

I still have been able to obtain the area I want, I continue,

Before I subtract, let me add an area as big as $$\displaystyle b²$$
next to the part I will be subtracting, this will give me an area that I can express easier $$\displaystyle (ab)$$

My identity now is as follows;

$$\displaystyle a²-ab+b²$$

We have subtracted $$\displaystyle ab$$ from  $$\displaystyle a²$$and added an area of $$\displaystyle b²$$.

Now, let’s cut the part that I need to subtract from the area including the area that I have added

My identity;

$$\displaystyle ab$$  area is also to be subtracted from my last identity;

$$\displaystyle a²-ab+b²-ab$$

If I organize

$$\displaystyle a²-2ab+b²$$

The final area that I have achieved;

$$\displaystyle a²-2ab+b²$$

## Identity of $$\displaystyle a²+b²$$

A square that has the length of $$\displaystyle b$$ ($$\displaystyle b²$$) is to be added to a square that has the length of $$\displaystyle a$$
(area $$\displaystyle a²$$)

I must find another way to express this area.

Let’s change the other length of this shape, change it to $$\displaystyle a+b$$ and turn it into a square.

I must subtract all white areas from the square $$\displaystyle (a+b)$$ .

$$\displaystyle (a+b).(a+b)-(ab+b²+ab-b²)$$
The total area of the square - The total area of the white areas

$$\displaystyle +b²$$ and $$\displaystyle -b²$$ simplifies each other.

$$\displaystyle (a+b).(a+b)-(ab+ab)$$

$$\displaystyle (a+b)²-2ab$$

Thus the identical is ;

$$\displaystyle a²+b²=(a+b)²-2ab$$

## Identity of $$\displaystyle a²-b²$$

When I subtract a $$\displaystyle b²$$ area from $$\displaystyle a²$$ , I must be able to express what is left behind in another way.

I must find another way to express this area.

I have cut the shape from the corner. I have obtained two equal trapezoids.

By turning one on its head, I have stuck it next to other one. I have obtained a rectangle.

The area of the rectangle :  $$\displaystyle (a-b).(a+b)$$

Thus;

$$\displaystyle a²- b²=(a-b).(a+b)$$

Let’s find this another way ;

The area is equal to the total area of these two rectangles.

$$\displaystyle a.(a-b) +b. (a-b)$$

Let me place the common multiplier in brackets, As $$\displaystyle (a-b)$$ is multiplied by both $$\displaystyle a$$ and $$\displaystyle b$$, it is the ‘common multiplier’.

$$\displaystyle a.(a-b) +b. (a-b) = ( a+ b) . ( a - b)$$

In earlier years, you had learnt the topic of distributive property of multiplication.

$$\displaystyle 3.(8+5) =3.8+3.5$$

We have been given the expression of 3.8 + 3.5 ; it’s as if we have placed the common  multiplier 3 within  parenthesis.

$$\displaystyle a²- b²= ( a+b) . (a -b )$$