This is a formula for calculating the 2th power of a number, for example if you want to calculate the 2th power of 5:

$\begin{align*} a&= 2\\ b&= 3 \end{align*}$

$ 5^2 = a^2 + b^2 + 2(ab)$

Sorry if that's not the right notation, but you'll get the point. Question is, what's the formula/method called?

  • $\begingroup$ That's a binomial expansion. $\endgroup$ – J. M. is a poor mathematician Sep 8 '11 at 17:42
  • 3
    $\begingroup$ Binomial expansions are $$(a+b)^1=a^1+b^1 $$ $$(a+b)^2=a^2+2ab+b^2$$ $$(a+b)^3=a^3+3a^2b+3ab^2+b^3$$ $$(a+b)^4=a^4+4a^3b+6a^2b^2+4ab^3+b^4$$ and so on. $\endgroup$ – anon Sep 8 '11 at 17:45
  • $\begingroup$ @anon, could you make an answer out of that so that I can confirm it as the best one;) $\endgroup$ – Jasper Kennis Sep 8 '11 at 18:08

As J. M. and lhf said, this is an example of a Binomial expansion. Other examples include $$(a+b)^1=a^1+b^1$$ $$(a+b)^2=a^2+2ab+b^2$$ $$(a+b)^3=a^3+3a^2b+3ab^2+b^3$$ $$(a+b)^4=a^4+4a^3b+6a^2b^2+4ab^3+b^4$$ and so on. The coefficient of the factor $a^m b^{n-m}$ in the expansion of $(a+b)^n$ is the number of ways of picking $m$ instances of $a$ out of $n$ possible instances of $a$ or $b$, which is the binomial coefficient $${n\choose m}=\frac{n!}{m!(n-m)!}.$$


It's just the binomial theorem for the second power. Or you can just expand $(a+b)^2= (a+b)\cdot(a+b)= a\cdot a + b \cdot a + a\cdot b + b \cdot b = a^2 + 2ab + b^2$.


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