Основные формулы по алгебре

c ÌàòÁþðî ðåøåíèå çàäà÷ ïî âûñøåé ìàòåìàòèêå, òåîðèè âåðîÿòíîñòåé
Îñíîâíûå ôîðìóëû ïî àëãåáðå
Ôîðìóëû ñîêðàùåííîãî óìíîæåíèÿ
(x − y)(x + y) = x2 − y 2 ,
(x + y)2 = x2 + 2xy + y 2 ,
x3 + y 3 = (x + y)(x2 − xy + y 2 ),
(x + y)3 = x3 + 3x2 y + 3xy 2 + y 3 ,
(x − y)2 = x2 − 2xy + y 2 .
x3 − y 3 = (x − y)(x2 + xy + y 2 )
(x − y)3 = x3 − 3x2 y + 3xy 2 − y 3 .
Êâàäðàòíîå óðàâíåíèå
Êîðíè êâàäðàòíîãî óðàâíåíèÿ ax2 + bx + c = 0 íàõîäÿò ïî ôîðìóëå
√
−b ± b2 − 4ac
.
x1,2 =
2a
Êîðíè êâàäðàòíîãî óðàâíåíèÿ ñ ÷åòíûì âòîðûì êîýôôèöèåíòîìax2 +2kx+c = 0 íàõîäÿò
ïî ôîðìóëå
√
−k ± k 2 − ac
x1,2 =
.
a
Ôîðìóëû Âèåòà äëÿ êîðíåé ïðèâåäåííîãî êâàäðàòíîãî óðàâíåíèÿ x2 + px + q = 0
x1 + x2 = −p,
x1 · x2 = q.
Ïðîñòåéøèå ñóììû
n(n + 1)
.
2
1 + 3 + 5 + · · · + (2n − 1) = n2 .
1 + 2 + 3 + ··· + n =
2 + 4 + 6 + · · · + 2n = n(n + 1).
n(n + 1)(2n + 1)
.
6
n(4n2 − 1)
12 + 32 + 52 + · · · + (2n − 1)2 =
.
3
1
1
1
n
+
+ ··· +
=
.
1·2 2·3
n(n + 1)
n+1
1
1
1
1 1
1
+
+ ··· +
=
−
.
1·2·3 2·3·4
n(n + 1)(n + 2)
2 2 (n + 1)(n + 2)
1 2 + 22 + 32 + · · · + n2 =