FlashMath | algebra

algebraic identities

a² + b² =
(a + b)² - 2ab
a² + b² =
(a - b)² + 2ab
(a - b)² =
(a + b)² - 4ab
(a + b + c)² =
a² + b² + c² + 2(ab + bc + ca)
(a + b)³ =
a³ + b³ + 3ab(a + b)
(a - b)³ =
a³ - b³ - 3ab(a - b)
a² + b² + c² - ab - bc - ca =
½ × [(a - b)² + (b - c)² + (c - a)²]
a³ + b³ + c³ - 3abc =
(a + b + c)(a² + b² + c² - ab - bc - ca)
a³ + b³ =
(a - b)³ + 3ab(a - b)
a² + b² + c² - ab - bc - ca = 0 implies =>
a = b = c
a³ + b³ =
(a + b)³ - 3ab(a + b)
a³ + b³ + c³ - 3abc = 0 implies =>
(a = b = c = 0) or (a = b = c)

logarithms

note: logab = c if ac (b > 0 and a > 0 and a ≠ 1)

loga(mn)
logam + logan
loga(m/n)
logam - logan
n logam
loga(mn)
logcb / logca
logab
N
alogaN
blogca
alogcb
logaa
1
loga1
0
1 / logab
logba
logad
logab × logbc × logcd
(1 / n) logab
loganb

polynomials

relations between roots and coefficients in a quadratic equation ax2 + bx + c

Quadratic Formula
x = -b ± √(b2 - 4ac) / 2a
α + β =
-b / a
α × β =
c / a
| α - β | =
√(b2 - 4ac) / | a |

relations between roots and coefficients in a cubic equation ax3 + bx2 + cx + d

α + β + γ =
-b / a
αβ + βγ + γα =
c / a
αβγ =
-d / a

newton's identity

if α and β are the roots of ax2 + bx + c and Sn = αⁿ ± βⁿ then
aSn + bSn-1 + cSn-2 = 0

Condition for ____ in a1x² + b1x + c1 = 0 and a2x² + b2x + c2 = 0

one roots common:
(c1a2 - c2a1)² = (b1c2 - b2c1)(a1b2 - a2b1)

both roots common:

a1/a2 = b1/b2 = c1/c2

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