# Does the fact that $x^2=(x-1)(x+1)+1$ have a name?

$$x^2 = (x-1)(x+1) +1$$

So:

\begin{align} 1^2 &= \phantom{1}0\cdot\phantom{1}2+1 = 1 \\ 2^2 &= \phantom{1}1\cdot\phantom{1}3+1 = 4 \\ 3^2 &= \phantom{1}2\cdot\phantom{1}4+1 = 9 \\ 4^2 &= \phantom{1}3\cdot\phantom{1}5+1 = 16 \\ 9^2 &= \phantom{1}8\cdot10+1 = 81 \\ 15^2 &= 14\cdot16+1 = 225 \end{align} and so on.

Then, to know any number raised to the power of $$2$$, you can multiply the previous number $$(x-1)$$ by the next one $$(x+1)$$, and add $$1$$.

So, to know the squared root of a number like $$64$$, you have to substract $$1$$ ($$63$$) and look for two numbers multiplied are $$63$$ and subtracted are $$2$$:

$$x \cdot y = 63 \qquad x - y = 2$$

Solving the equation you get $$9$$ and $$7$$ (or $$-7$$ and $$-9$$). The number between these is the square root ($$8$$).

I don't know if this apply for power of $$3$$.

Does this fact/theorem/relation has a name or something?

• Here is a link to help with the MathJax notation math.stackexchange.com/help/notation
– user475040
Sep 21, 2018 at 19:44
• $(x-1)(x+1)=x^2-1$ is what I think you are trying to write.
– lulu
Sep 21, 2018 at 19:50
• You may like $x^3=(x-1)x(x+1)+x$
– N74
Sep 22, 2018 at 7:56

The equation in your pattern is $$x^2 = (x-1)(x+1) + 1$$.

Both sides of the equation evaluate to $$x^2$$. It's just written differently on the right side. Expand the right side (using FOIL) and you get

$$(x-1)(x+1) + 1$$ $$= x^2 - x + x -1 + 1$$ $$= x^2$$

This is basic algebra. No special theorems are involved.

If you were to subtract 1 from both sides, you would get $$x^2 - 1 = (x + 1)(x - 1)$$ This is a simpler case of the more general factoring of the difference of squares: $$x^2 - b^2 = (x + b)(x - b)$$

A similar formula for $$x^3$$ is $$x^3=(x-1)(x^2+x+1)+1$$

You can extend this to higher powers

$$x^n= (x-1)(x^{n-1}+x^{n-2}+...+1)+1$$ Which is the basis for the sum of geometric series formula.

$$1+x+x^2+...+x^{n} = \frac {x^{n+1}-1}{x-1}$$

Trickery;

$$x^2= x^2 -1 +1 = (x^2-1) +1=$$

$$(x+1)(x-1)+1$$.

Used: $$x^2-1=(x+1)(x-1)$$.

P.S. Example :

$$19^2= (20)(18)+1 = 360 +1=$$

$$361$$.