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Beliebt Algebra >

x^2-8x+15=(ax+b)(cx+d)

Lösung

x^{2} - 8 x + 15 = (a x + b) (c x + d)

Lösung

x = \frac{8 + b c + a d + a ^{2} d ^{2} + 60 a c + 16 a d - 2 ab c d + b ^{2} c ^{2} + 16 b c + 4 b d + 4}{2 ( 1 - a c )}, x = \frac{8 + b c + a d - a ^{2} d ^{2} + 60 a c + 16 a d - 2 ab c d + b ^{2} c ^{2} + 16 b c + 4 b d + 4}{2 ( 1 - a c )}; 1 - a c \neq = 0

Schritte zur Lösung

x^{2} - 8 x + 15 = (a x + b) (c x + d)

Schreibe

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

um:

a c x^{2} + a d x + b c x + b d

x^{2} - 8 x + 15 = a c x^{2} + a d x + b c x + b d

Subtrahiere

b c x + b d

von beiden Seiten

x^{2} - 8 x + 15 - (b c x + b d) = a c x^{2} + a d x + b c x + b d - (b c x + b d)

Vereinfache

x^{2} - 8 x + 15 - b c x - b d = a c x^{2} + a d x

Verschiebe

a d x

auf die linke Seite

x^{2} - 8 x + 15 - b c x - b d - a d x = a c x^{2}

Verschiebe

a c x^{2}

auf die linke Seite

x^{2} - 8 x + 15 - b c x - b d - a d x - a c x^{2} = 0

Schreibe in der Standard Form

a x^{2} + b x + c = 0

(1 - a c) x^{2} - (8 + b c + a d) x + 15 - b d = 0

Löse mit der quadratischen Formel

x_{1, 2} = \frac{- ( - 8 - b c - a d ) \pm ( - 8 - b c - a d ) ^{2} - 4 ( 1 - a c ) ( 15 - b d )}{2 ( 1 - a c )}

Vereinfache

(- 8 - b c - a d)^{2} - 4 (1 - a c) (15 - b d) : a^{2} d^{2} + 60 a c + 16 a d - 2 ab c d + b^{2} c^{2} + 16 b c + 4 b d + 4

x_{1, 2} = \frac{- ( - 8 - b c - a d ) \pm a ^{2} d ^{2} + 60 a c + 16 a d - 2 ab c d + b ^{2} c ^{2} + 16 b c + 4 b d + 4}{2 ( 1 - a c )}; 1 - a c \neq = 0

Trenne die Lösungen

x_{1} = \frac{- ( - 8 - b c - a d ) + a ^{2} d ^{2} + 60 a c + 16 a d - 2 ab c d + b ^{2} c ^{2} + 16 b c + 4 b d + 4}{2 ( 1 - a c )}, x_{2} = \frac{- ( - 8 - b c - a d ) - a ^{2} d ^{2} + 60 a c + 16 a d - 2 ab c d + b ^{2} c ^{2} + 16 b c + 4 b d + 4}{2 ( 1 - a c )}

x = \frac{- ( - 8 - b c - a d ) + a ^{2} d ^{2} + 60 a c + 16 a d - 2 ab c d + b ^{2} c ^{2} + 16 b c + 4 b d + 4}{2 ( 1 - a c )} : \frac{8 + b c + a d + a ^{2} d ^{2} + 60 a c + 16 a d - 2 ab c d + b ^{2} c ^{2} + 16 b c + 4 b d + 4}{2 ( 1 - a c )}

x = \frac{- ( - 8 - b c - a d ) - a ^{2} d ^{2} + 60 a c + 16 a d - 2 ab c d + b ^{2} c ^{2} + 16 b c + 4 b d + 4}{2 ( 1 - a c )} : \frac{8 + b c + a d - a ^{2} d ^{2} + 60 a c + 16 a d - 2 ab c d + b ^{2} c ^{2} + 16 b c + 4 b d + 4}{2 ( 1 - a c )}

Die Lösungen für die quadratische Gleichung sind:

x = \frac{8 + b c + a d + a ^{2} d ^{2} + 60 a c + 16 a d - 2 ab c d + b ^{2} c ^{2} + 16 b c + 4 b d + 4}{2 ( 1 - a c )}, x = \frac{8 + b c + a d - a ^{2} d ^{2} + 60 a c + 16 a d - 2 ab c d + b ^{2} c ^{2} + 16 b c + 4 b d + 4}{2 ( 1 - a c )}; 1 - a c \neq = 0

Graph

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