Formula about square
\(a^2-b^2=(a+b)(a-b)\)
\((x+a)(x+b)=x^2+(a+b)x+ab\)
\(a^2+b^2=(a+b)^2-2ab\)
\(a^2+b^2=(a-b)^2+2ab\)
\((a+b)^2=(a-b)^2+4ab\)
\((a-b)^2=(a+b)^2-4ab\)
\(a^2+b^2=\frac{(a+b)+(a-b)}{2}\)
\(ab=(\frac{a+b}{2})^2-(\frac{a-b}{2})^2\)
\((a+b+c)^2=a^3+b^2+c^2+2(ab+bc+ac)\)
\(a^2+b^2+c^2=(a+b+c)^2-2(ab+bc+ac)\)
\(2(ab+bc+ac)=(a+b+c)^2-(a^2+b^2+c^2)\)
\((a+b-c)^2=a^2+b^2+c^2+2(ab-bc-ac)\)
\((a-b+c)^2=a^2+b^2+c^2+2(ac-ab-bc)\)
\((a-b-c)^2=a^2+b^2+c^2-2(ab-2bc+2ac)\)
Formula about cube
\((a+b)^3=a^3+3a^2b+3ab^2+b^3\)
\(=a^3+b^3+3ab(a+b)\)
\(a^3+b^3=(a+b)^3-3ab(a+b)\)
\((a-b)^3=a^3-3a^2b+3ab^2-b^3\)
=\(a^3-b^3-3ab(a-b)\)
\(a^3-b^3=(a-b)^3+3ab(a-b)\)
\(a^3+b^3)=(a+b)(a^2-ab+b^2)\)
\(a^3-b^3)=(a-b)(a^2+ab+b^2)\)
Product formulas
\((a-b)^2=a^2-2ab+b^2\)
\((a+b)^2=a^2+2ab+b^2\)
\((a-b)^3=a^3-3a^2b+3ab^2-b^3\)
\((a+b)^3=a^3+3a^2b+3ab^2+b^3\)
\((a-b)^4=a^4-4a^3b+6a^2b^2-4ab^3+b^4\)
(\(a+b)^4=a^4+4a^b+6a^2b^2+4ab^3+b^4\)
Factoring formula
\(a^2-b^2=(a+b)(a-b)\)
\(a^3-b^3=(a-b)(a^2+ab+b^2)\)
\(a^3+b^3=(a+b)(a^2-ab+b^2)\)
\(a^4-b^4=(a^2-b^2)(a^2+b^2)\)
=\((a-b)(a+b)(a^2+b^2)\)
\(a^5-b^5=(a-b)(a^4+a^3b+a^2b^2+ab^3+b^4)\)
\(a^n+b^n=(a+b)(a^{n-1} -a^{n-2}b+a^{n-3}b^2-......-ab^{n-2}+b^{n-1}\)
When n is even number↓
\(a^n-b^n= (a-b)(a^{n-1}+a^{n-2}b+a^{n-3}b^2+......+ab^{n-2}+b^{n-1}\)
\(a^n+b^n=(a+b)(a^{n-1} -a^{n-2}b+a^{n-3}b^2-......+ab^{n-2}+b^{n-1}\)
Power and logarithm
\(a^n=a×a×a×a×......×a\)
\(a^0=1\)
\(a^{-n}=\frac{1}{a^n}\)
\(\sqrt{a}=a^{\frac{1}{2}}\)
\(\sqrt[n]{a}=a^{\frac{1}{n}}\)
\(\sqrt[n]{a^m}=(\sqrt[n]{a})^m=a^{\frac{m}{n}}\)
\(a^{(m)n}=a^{mn}\)
\((ab)^m=a^mb^m\)
\((\frac{a}{b})^m=\frac{a^m}{b^m}\)
\(a^m.a^n=a^{m+n}\)
\(\frac{a^m}{a^n}=a^{m-n}\)
If \(a^x=a^y → x=y\)
If\(a^m=b^m→ a=b\)
If \(a^x=n → x=log_an\)
If \(x=log_an→ a^x=n\)
\(log1=0\)
\(log0=undefined\)
\(log_aa=1\)
\(log_ab=\frac{1}{log_ba}\)
\(log_aM=log_bM×log_ab\)
\(log_aM=\frac{log_bM}{log_ba}\)
\(log_aM^r=rlog_aM\)
\(log_a(MN)=log_aM+log_aN\)
\(log_a\frac{M}{N}=log_aM-log_aN\)
\(log_a\sqrt[n]{m}=\frac{1}{n}log_am\)
\(a^{log_xb}=b^{log_xa}\)
\(x^y=e^{ylog_ex}\)
(Series/progression)
The arithmetic series
Ideal arithmetic series=a+(a+d)+(a+2d)+(a+3d)+...+n
First term=a,
Number of terms=n,
Common difference=second term-first term,
The nth term of series=a+(n-1)d,
N =\(\frac{nth term-1}{d} -1\)
Sum of n term=\(\frac{n}{2}[{2a+(n-1)d}]\)
Proportional series
Ideal proportional series=\(a+ar+ar^2+ar^3+......+ar^{n-1}\)
First term=a.
Common proportion=\(r=\frac{second term}{first term}\)
Number of term=n,
nth term=\(ar^{n-1} ;when, r>1\)
Sum of n terms=\(a×\frac{r^{n-1}}{r-1}\)
When \(r<1\)
Sum of n terms=\(a×\frac{1-r^n}{1-r}\)