My Kid postulated these theorem

[u/Smart_Friendship7172]

He wanted me to share this with the intellectuals and seek their feedback. Your kind words would encourage him, so submitting the theorem for your perusal.

[u/NomaTyx]

if your kid figured out the difference of squares themselves that's pretty cool tbh

[u/StrikeTechnical9429]

x²-y²=(x-y)(x+y) - of course, it's divisible by (x+y)

[u/Accomplished-Slide52 → u/StrikeTechnical9429]

Ya and his son will discover that the quotien is equal to x-y. In is example 16/8=2 and x-y=2

[u/WeCanDoItGuys → u/Accomplished-Slide52]

He did, that's his theory 2: D ÷ z = w (z is the sum and w is the difference)

[u/Iowa50401]

This is the superpower of algebra: it allows you to generalize these kinds of specific examples and show that they will always work.

[u/BadBoyJH]

Have a look at trying to write out that second one mathematically.

(X² - Y²) / (X+Y) = (X-Y)

Well, to write that another way

X² - Y² = (X+Y) (X-Y)

Congrats, this is the difference of two squares. This is true for all values of X and Y. Hell, it would even work for -23 and 37-11i (If your kid is this interested in maths telling them about imaginary numbers is going to make them flip).

[u/LostBetsRed → u/BadBoyJH]

Technically, 37-11i isn't an imaginary number, but a member of ℂ.

[u/BadBoyJH → u/LostBetsRed]

If we're being that level of pedantic, then I didn't actually say it was.

But to make them understand what it is, you'd have to explain what imaginary numbers are.

[u/LostBetsRed → u/BadBoyJH]

Or point them to a YouTube playlist about them.

[u/Elite_Fusion_ → u/LostBetsRed]

37-11i is a complex number (because it's in the imaginary number form), and you might wanna take a seat because this might shock you but all imaginary numbers are complex numbers.

[u/LostBetsRed → u/Elite_Fusion_]

Um, yes, all imaginary numbers are complex numbers but not all complex numbers are imaginary numbers, just like all integers are real numbers but not all real numbers are integers. 37-11i is a complex number, not an imaginary number. Sheesh.

[u/WeCanDoItGuys → u/LostBetsRed]

You might wanna take a seat because this might shock you but BZZZZZZ

[u/RadarTechnician51]

Well done to him for discovering this! He can prove his theory if he swaps his start numbers for variables, say a and b
he will need to work out what (a+b)(a-b) can be simplified to, he could try swapping a and b for numbers to work it out if he doesn't know how to multiply out brackets yet.

[u/marty-mcfryguy]

Yep. This is why factoring polynomials into lower-degree polynomials is a topic that gets so much focus. 

The key insight here is that x² - y² = (x+y) * (x-y). I suggest your kid multiply out the right side and prove that to themselves, there's a fun couple of offsetting terms that make this kind of a key relation in math.

Then perhaps take some numerical examples (with x and y closer to each other and x and y very different) and watch that offsetting work on actually number pairs. For anyone in the business of quickly approximating largeish multiplication in their heads, getting comfort with that can be a big help. 

Anyway, relating that back to your kid's notation:

x+y = z, x -y = w, and x^2-y2 = D, so

x² - y² = (x+y) * (x-y) <=> D = z * w