| source: knowledge_base | |
| topic: algebra | |
| class_level: class_10 | |
| difficulty: intermediate | |
| Algebra and Number Theory: | |
| Quadratic: x=(-b±sqrt(b^2-4ac))/2a. D=b^2-4ac. | |
| D>0: two real roots. D=0: repeated root. D<0: complex roots. | |
| log(ab)=log(a)+log(b). log(a^n)=n*log(a). | |
| a^2-b^2=(a+b)(a-b). a^3+b^3=(a+b)(a^2-ab+b^2). | |
| AP: S_n=n/2*(2a+(n-1)d). GP: S=a(1-r^n)/(1-r). Infinite GP: a/(1-r) for |r|<1. |