(m^2-n^2)x^2-4mnx=m^2-n^2

Lösung

(m^{2} - n^{2}) x^{2} - 4 mn x = m^{2} - n^{2}

x = \frac{2 mn + 4 m ^{2} n ^{2} - ( m ^{2} - n ^{2} ) ( n ^{2} - m ^{2} ) ^{2}}{m ^{2} - n ^{2}}, x = \frac{2 mn - 4 m ^{2} n ^{2} - ( m ^{2} - n ^{2} ) ( n ^{2} - m ^{2} ) ^{2}}{m ^{2} - n ^{2}}; m^{2} - n^{2} \neq = 0

Schritte zur Lösung

(m^{2} - n^{2}) x^{2} - 4 mn x = m^{2} - n^{2}

Verschiebe

n^{2}

auf die linke Seite

(m^{2} - n^{2}) x^{2} - 4 mn x + n^{2} = m^{2}

Verschiebe

m^{2}

auf die linke Seite

(m^{2} - n^{2}) x^{2} - 4 mn x + n^{2} - m^{2} = 0

Löse mit der quadratischen Formel

x_{1, 2} = \frac{- ( - 4 mn ) \pm ( - 4 mn ) ^{2} - 4 ( m ^{2} - n ^{2} ) ( n ^{2} - m ^{2} )}{2 ( m ^{2} - n ^{2} )}

Vereinfache

(- 4 mn)^{2} - 4 (m^{2} - n^{2}) (n^{2} - m^{2}) : 2 4 m^{2} n^{2} - (m^{2} - n^{2}) (- m^{2} + n^{2})^{2}

x_{1, 2} = \frac{- ( - 4 mn ) \pm 2 4 m ^{2} n ^{2} - ( m ^{2} - n ^{2} ) ( - m ^{2} + n ^{2} ) ^{2}}{2 ( m ^{2} - n ^{2} )}; m^{2} - n^{2} \neq = 0

Trenne die Lösungen

x_{1} = \frac{- ( - 4 mn ) + 2 4 m ^{2} n ^{2} - ( m ^{2} - n ^{2} ) ( - m ^{2} + n ^{2} ) ^{2}}{2 ( m ^{2} - n ^{2} )}, x_{2} = \frac{- ( - 4 mn ) - 2 4 m ^{2} n ^{2} - ( m ^{2} - n ^{2} ) ( - m ^{2} + n ^{2} ) ^{2}}{2 ( m ^{2} - n ^{2} )}

x = \frac{- ( - 4 mn ) + 2 4 m ^{2} n ^{2} - ( m ^{2} - n ^{2} ) ( - m ^{2} + n ^{2} ) ^{2}}{2 ( m ^{2} - n ^{2} )} : \frac{2 mn + 4 m ^{2} n ^{2} - ( m ^{2} - n ^{2} ) ( n ^{2} - m ^{2} ) ^{2}}{m ^{2} - n ^{2}}

x = \frac{- ( - 4 mn ) - 2 4 m ^{2} n ^{2} - ( m ^{2} - n ^{2} ) ( - m ^{2} + n ^{2} ) ^{2}}{2 ( m ^{2} - n ^{2} )} : \frac{2 mn - 4 m ^{2} n ^{2} - ( m ^{2} - n ^{2} ) ( n ^{2} - m ^{2} ) ^{2}}{m ^{2} - n ^{2}}

Die Lösungen für die quadratische Gleichung sind:

x = \frac{2 mn + 4 m ^{2} n ^{2} - ( m ^{2} - n ^{2} ) ( n ^{2} - m ^{2} ) ^{2}}{m ^{2} - n ^{2}}, x = \frac{2 mn - 4 m ^{2} n ^{2} - ( m ^{2} - n ^{2} ) ( n ^{2} - m ^{2} ) ^{2}}{m ^{2} - n ^{2}}; m^{2} - n^{2} \neq = 0

- 3 x^{2} + x + 1 = 0

- w^{2} + 6 w + 72 = 0

so l v e f or x, x^{2} - 3 x y + 5 = 0

- 4 x^{2} - 80 = 0

0.0813600000000 \dots + 0.2431900000000 \dots \cdot x + 0.3564800000000 \dots \cdot x^{2} = 0