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العربية

((sqrt(x-1)-b-c))/a+((sqrt(x-1)-c-a))/b+((sqrt(x-1)-a-b))/c =3

Lösung

\frac{( x - 1 - b - c )}{a} + \frac{( x - 1 - c - a )}{b} + \frac{( x - 1 - a - b )}{c} = 3

x = \frac{( 3 b c a + b ^{2} c + b c ^{2} + c ^{2} a + c a ^{2} + b a ^{2} + b ^{2} a ) ^{2}}{b ^{2} c ^{2} + 2 b c ^{2} a + 2 b ^{2} c a + 2 b c a ^{2} + b ^{2} a ^{2} + c ^{2} a ^{2}} + 1

Schritte zur Lösung

\frac{( x - 1 - b - c )}{a} + \frac{( x - 1 - c - a )}{b} + \frac{( x - 1 - a - b )}{c} = 3

Finde das kleinste gemeinsame Vielfache von

a, b, c : b c a

Multipliziere mit dem kleinsten gemeinsamen Multiplikator=

b c a

\frac{x - 1 - b - c}{a} b c a + \frac{x - 1 - c - a}{b} b c a + \frac{x - 1 - a - b}{c} b c a = 3 b c a

Vereinfache

b c (x - 1 - b - c) + c a (x - 1 - c - a) + ba (x - 1 - a - b) = 3 b c a

Schreibe

b c (x - 1 - b - c) + c a (x - 1 - c - a) + ba (x - 1 - a - b)

um:

b c x - 1 - b^{2} c - b c^{2} + c a x - 1 - c^{2} a - c a^{2} + ba x - 1 - b a^{2} - b^{2} a

b c x - 1 - b^{2} c - b c^{2} + c a x - 1 - c^{2} a - c a^{2} + ba x - 1 - b a^{2} - b^{2} a = 3 b c a

Verschiebe

b^{2} c

auf die rechte Seite

b c x - 1 - b c^{2} + c a x - 1 - c^{2} a - c a^{2} + ba x - 1 - b a^{2} - b^{2} a = 3 b c a + b^{2} c

Verschiebe

b c^{2}

auf die rechte Seite

b c x - 1 + c a x - 1 - c^{2} a - c a^{2} + ba x - 1 - b a^{2} - b^{2} a = 3 b c a + b^{2} c + b c^{2}

Verschiebe

c^{2} a

auf die rechte Seite

b c x - 1 + c a x - 1 - c a^{2} + ba x - 1 - b a^{2} - b^{2} a = 3 b c a + b^{2} c + b c^{2} + c^{2} a

Verschiebe

c a^{2}

auf die rechte Seite

b c x - 1 + c a x - 1 + ba x - 1 - b a^{2} - b^{2} a = 3 b c a + b^{2} c + b c^{2} + c^{2} a + c a^{2}

Verschiebe

b a^{2}

auf die rechte Seite

b c x - 1 + c a x - 1 + ba x - 1 - b^{2} a = 3 b c a + b^{2} c + b c^{2} + c^{2} a + c a^{2} + b a^{2}

Verschiebe

b^{2} a

auf die rechte Seite

b c x - 1 + c a x - 1 + ba x - 1 = 3 b c a + b^{2} c + b c^{2} + c^{2} a + c a^{2} + b a^{2} + b^{2} a

Faktorisiere

b c x - 1 + c a x - 1 + ba x - 1 : x - 1 (b c + c a + ba)

x - 1 (b c + c a + ba) = 3 b c a + b^{2} c + b c^{2} + c^{2} a + c a^{2} + b a^{2} + b^{2} a

Teile beide Seiten durch

b c + c a + ba

x - 1 = \frac{3 b c a + b ^{2} c + b c ^{2} + c ^{2} a + c a ^{2} + b a ^{2} + b ^{2} a}{b c + c a + ba}

Quadriere beide Seiten:

x - 1 = \frac{( 3 b c a + b ^{2} c + b c ^{2} + c ^{2} a + c a ^{2} + b a ^{2} + b ^{2} a ) ^{2}}{b ^{2} c ^{2} + 2 b c ^{2} a + 2 b ^{2} c a + 2 b c a ^{2} + b ^{2} a ^{2} + c ^{2} a ^{2}}

x - 1 = \frac{( 3 b c a + b ^{2} c + b c ^{2} + c ^{2} a + c a ^{2} + b a ^{2} + b ^{2} a ) ^{2}}{b ^{2} c ^{2} + 2 b c ^{2} a + 2 b ^{2} c a + 2 b c a ^{2} + b ^{2} a ^{2} + c ^{2} a ^{2}}

Löse

x - 1 = \frac{( 3 b c a + b ^{2} c + b c ^{2} + c ^{2} a + c a ^{2} + b a ^{2} + b ^{2} a ) ^{2}}{b ^{2} c ^{2} + 2 b c ^{2} a + 2 b ^{2} c a + 2 b c a ^{2} + b ^{2} a ^{2} + c ^{2} a ^{2}} : x = \frac{( 3 b c a + b ^{2} c + b c ^{2} + c ^{2} a + c a ^{2} + b a ^{2} + b ^{2} a ) ^{2}}{b ^{2} c ^{2} + 2 b c ^{2} a + 2 b ^{2} c a + 2 b c a ^{2} + b ^{2} a ^{2} + c ^{2} a ^{2}} + 1

x = \frac{( 3 b c a + b ^{2} c + b c ^{2} + c ^{2} a + c a ^{2} + b a ^{2} + b ^{2} a ) ^{2}}{b ^{2} c ^{2} + 2 b c ^{2} a + 2 b ^{2} c a + 2 b c a ^{2} + b ^{2} a ^{2} + c ^{2} a ^{2}} + 1

2 5 - 4 x = - 12

4 x - 3 + 2 x + 3 = 6

3 2 x + 4 2 x = 10 + 2 2 x

4 x - 8 - 1 = 2 x - 5

x + 7 + x - 1 - 2 x + 2 = 0