2x^3-8x^2-24x=ax(x+b)(x+c)

Lösung

2 x^{3} - 8 x^{2} - 24 x = a x (x + b) (x + c)

x = 0, x = \frac{8 + a c + ab + a ^{2} b ^{2} + a ^{2} c ^{2} - 2 a ^{2} b c + 16 ab + 16 a c + 8 ab c - 96 a + 256}{2 ( 2 - a )}, x = \frac{8 + a c + ab - a ^{2} b ^{2} + a ^{2} c ^{2} - 2 a ^{2} b c + 16 ab + 16 a c + 8 ab c - 96 a + 256}{2 ( 2 - a )}; a \neq = 2

Schritte zur Lösung

2 x^{3} - 8 x^{2} - 24 x = a x (x + b) (x + c)

Verschiebe

a x (x + b) (x + c)

auf die linke Seite

2 x^{3} - 8 x^{2} - 24 x - a x (x + b) (x + c) = 0

Faktorisiere

2 x^{3} - 8 x^{2} - 24 x - a x (x + b) (x + c) : x (2 x^{2} - 8 x - a x^{2} - a c x - ab x - ab c - 24)

x (2 x^{2} - 8 x - a x^{2} - a c x - ab x - ab c - 24) = 0

Anwendung des Nullfaktorprinzips: Wenn

ab = 0

dann

a = 0

oder

b = 0

x = 0 or 2 x^{2} - 8 x - a x^{2} - a c x - ab x - ab c - 24 = 0

Löse

2 x^{2} - 8 x - a x^{2} - a c x - ab x - ab c - 24 = 0 : x = \frac{8 + a c + ab + a ^{2} b ^{2} + a ^{2} c ^{2} - 2 a ^{2} b c + 16 ab + 16 a c + 8 ab c - 96 a + 256}{2 ( 2 - a )}, x = \frac{8 + a c + ab - a ^{2} b ^{2} + a ^{2} c ^{2} - 2 a ^{2} b c + 16 ab + 16 a c + 8 ab c - 96 a + 256}{2 ( 2 - a )}; a \neq = 2

Die Lösungen sind

x = 0, x = \frac{8 + a c + ab + a ^{2} b ^{2} + a ^{2} c ^{2} - 2 a ^{2} b c + 16 ab + 16 a c + 8 ab c - 96 a + 256}{2 ( 2 - a )}, x = \frac{8 + a c + ab - a ^{2} b ^{2} + a ^{2} c ^{2} - 2 a ^{2} b c + 16 ab + 16 a c + 8 ab c - 96 a + 256}{2 ( 2 - a )}; a \neq = 2

- 2 x^{3} = 32 x

27 x^{3} - 1 = 0

4 x^{2} - 9 = (2 x^{3}) (2 x - 3)

8 x^{3} = (- 2 x) (A)

9 x^{3} - 18 x^{2} - 4 x + 8 = 0