The Square of the Sum Formula and real world scenarios

[DS2C]

I'm starting with my self studying of math with Algebra I. The text I'm using is Gelfand and Shen's Algebra.
I'm at the point where it talks about the Formula for the Square of the Sum, The Square of the Distance Formula,
and The Difference of Squares Formula.

In going over this, I understand that the formula for the square of the sum is essentially saying the following:
If a boys each received a candies, they would walk away with a total of $a^2$ candies.
If b girls each received b candies, they would walk away with a total of $b^2$ candies.
In using this same scenario, but the boys received ab candies and the girls received ba candies, they would walk away with 2ab more than the previous scenarios. This equation is listed below.

$$\left(a+b\right)^2~=~a^2+b^2+2ab$$

Now I understand this logic, however in looking at the other two formulas, $\left(a-b\right)^2~=~a^2-2ab+b^2$ (the square of the distance formula) as well as $a^2-b^2~=~\left(a+b\right)\left(a-b\right)$ (the difference of squares formula), I can't put together a scenario in which these could be used in a scenario like in the first example.

Could someone give me some insight into what these last two are exactly use for?

 

[fresh_42]

$(a-b)^2=a^2-2ab+b^2$ is exactly the same as $(a+b)^2=a^2+2ab+b^2\,$, just with a negative $b$.
You can use all three to shortcut calculations in mind, because they only describe, what people do anyway:
$29^2 = 29 \cdot (20 +9) = 29 \cdot 20 + 29 \cdot 9 = 20 \cdot 20 + 9 \cdot 20 + 20 \cdot 9 + 9 \cdot 9 = 20^2+2\cdot20\cdot9+9^2=400+360+81=841$ and shorter with the right formula $29^2=(30-1)^2=900-2\cdot1\cdot30+1=841$.
Similar is true for the third formula:
$49 \cdot 51 = 2499$ because $(a-b)\cdot(a+b)=a^2 - b^2$ immediately gives $49\cdot51=(50-1)\cdot(50+1)=50^2-1\,.$

So basically these formulas save time: no need for written long multiplications or change of media by taking a calculator. Of course this is only the practical aspect of it. They are the surface of some deeper mathematical truths. You can a search for Pascal's triangle, e.g. here https://en.wikipedia.org/wiki/Pascal's_triangle and see aspects of combinatorics. And especially the third one, $(a-b)\cdot(a+b)=a^2 - b^2\,,$ is often used in the case of $b=1$.

They are simply a different way to write the same number, and if we talk about factorization of numbers, as is often the case in various mathematical areas, they come in handy. It's like Pythagoras' theorem: simple, true, easy to use and faster than complicated calculations.

 

[DS2C]

That was a great help, thank you for writing that out.

 

[Svein]

Could someone give me some insight into what these last two are exactly use for?

How to calculate the next square: If you know $x^{2}$, then $(x+1)^{2}=x^{2}+2\cdot x + 1$. It also shows that the difference between two successive integer squares is always an odd number.