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Beliebt Algebra >

solvefor y,(x^2+y^2+ax)^2=a^2(x^2+y^2)

Lösung

löse nach

y, (x^{2} + y^{2} + a x)^{2} = a^{2} (x^{2} + y^{2})

Lösung

y = \frac{- 2 x ^{2} - 2 a x + a ^{2} + - 4 a ^{3} x + a ^{4}}{2}, y = - \frac{- 2 x ^{2} - 2 a x + a ^{2} + - 4 a ^{3} x + a ^{4}}{2}, y = \frac{- 2 x ^{2} - 2 a x + a ^{2} - - 4 a ^{3} x + a ^{4}}{2}, y = - \frac{- 2 x ^{2} - 2 a x + a ^{2} - - 4 a ^{3} x + a ^{4}}{2}

Schritte zur Lösung

(x^{2} + y^{2} + a x)^{2} = a^{2} (x^{2} + y^{2})

Schreibe

(x^{2} + y^{2} + a x)^{2}

um:

2 a x^{3} + 2 y^{2} x^{2} + x^{4} + a^{2} x^{2} + 2 a y^{2} x + y^{4}

Schreibe

a^{2} (x^{2} + y^{2})

um:

a^{2} x^{2} + a^{2} y^{2}

2 a x^{3} + 2 y^{2} x^{2} + x^{4} + a^{2} x^{2} + 2 a y^{2} x + y^{4} = a^{2} x^{2} + a^{2} y^{2}

Subtrahiere

a^{2} x^{2} + a^{2} y^{2}

von beiden Seiten

2 a x^{3} + 2 y^{2} x^{2} + x^{4} + a^{2} x^{2} + 2 a y^{2} x + y^{4} - (a^{2} x^{2} + a^{2} y^{2}) = a^{2} x^{2} + a^{2} y^{2} - (a^{2} x^{2} + a^{2} y^{2})

Vereinfache

y^{4} + (2 x^{2} + 2 a x - a^{2}) y^{2} + x^{4} + 2 a x^{3} = 0

Schreibe die Gleichung um mit

u = y^{2}

und

u^{2} = y^{4}

u^{2} + (2 x^{2} + 2 a x - a^{2}) u + x^{4} + 2 a x^{3} = 0

Löse

u^{2} + (2 x^{2} + 2 a x - a^{2}) u + x^{4} + 2 a x^{3} = 0 : u = \frac{- 2 x ^{2} - 2 a x + a ^{2} + - 4 a ^{3} x + a ^{4}}{2}, u = \frac{- 2 x ^{2} - 2 a x + a ^{2} - - 4 a ^{3} x + a ^{4}}{2}

u = \frac{- 2 x ^{2} - 2 a x + a ^{2} + - 4 a ^{3} x + a ^{4}}{2}, u = \frac{- 2 x ^{2} - 2 a x + a ^{2} - - 4 a ^{3} x + a ^{4}}{2}

Setze

u = y^{2} w i e d er e in,

löse für

y

Löse

y^{2} = \frac{- 2 x ^{2} - 2 a x + a ^{2} + - 4 a ^{3} x + a ^{4}}{2} : y = \frac{- 2 x ^{2} - 2 a x + a ^{2} + - 4 a ^{3} x + a ^{4}}{2}, y = - \frac{- 2 x ^{2} - 2 a x + a ^{2} + - 4 a ^{3} x + a ^{4}}{2}

Löse

y^{2} = \frac{- 2 x ^{2} - 2 a x + a ^{2} - - 4 a ^{3} x + a ^{4}}{2} : y = \frac{- 2 x ^{2} - 2 a x + a ^{2} - - 4 a ^{3} x + a ^{4}}{2}, y = - \frac{- 2 x ^{2} - 2 a x + a ^{2} - - 4 a ^{3} x + a ^{4}}{2}

Die Lösungen sind

y = \frac{- 2 x ^{2} - 2 a x + a ^{2} + - 4 a ^{3} x + a ^{4}}{2}, y = - \frac{- 2 x ^{2} - 2 a x + a ^{2} + - 4 a ^{3} x + a ^{4}}{2}, y = \frac{- 2 x ^{2} - 2 a x + a ^{2} - - 4 a ^{3} x + a ^{4}}{2}, y = - \frac{- 2 x ^{2} - 2 a x + a ^{2} - - 4 a ^{3} x + a ^{4}}{2}

Graph

Beliebte Beispiele

x^{3} + 24 x + 16 = 0

x^3=38+17sqrt(5)

x^{3} = 38 + 175

(2 x^{2} - 4)^{2} + 9 = 0

1 - (4 - 3 x)^{3} = 9

16x^4-40x^2+9=0

16 x^{4} - 40 x^{2} + 9 = 0