mnx^2-(m^2-n^2)x+mn=0

Lösung

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

x = \frac{m ^{2} - n ^{2} + m ^{4} - 6 m ^{2} n ^{2} + n ^{4}}{2 mn}, x = \frac{m ^{2} - n ^{2} - m ^{4} - 6 m ^{2} n ^{2} + n ^{4}}{2 mn}; mn \neq = 0

Schritte zur Lösung

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

Schreibe

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

um:

mn x^{2} - m^{2} x + n^{2} x + mn

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

Schreibe in der Standard Form

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

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

Löse mit der quadratischen Formel

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

Vereinfache

(- m^{2} + n^{2})^{2} - 4 mnmn : m^{4} - 6 m^{2} n^{2} + n^{4}

x_{1, 2} = \frac{- ( - m ^{2} + n ^{2} ) \pm m ^{4} - 6 m ^{2} n ^{2} + n ^{4}}{2 mn}; mn \neq = 0

Trenne die Lösungen

x_{1} = \frac{- ( - m ^{2} + n ^{2} ) + m ^{4} - 6 m ^{2} n ^{2} + n ^{4}}{2 mn}, x_{2} = \frac{- ( - m ^{2} + n ^{2} ) - m ^{4} - 6 m ^{2} n ^{2} + n ^{4}}{2 mn}

x = \frac{- ( - m ^{2} + n ^{2} ) + m ^{4} - 6 m ^{2} n ^{2} + n ^{4}}{2 mn} : \frac{m ^{2} - n ^{2} + m ^{4} - 6 m ^{2} n ^{2} + n ^{4}}{2 mn}

x = \frac{- ( - m ^{2} + n ^{2} ) - m ^{4} - 6 m ^{2} n ^{2} + n ^{4}}{2 mn} : \frac{m ^{2} - n ^{2} - m ^{4} - 6 m ^{2} n ^{2} + n ^{4}}{2 mn}

Die Lösungen für die quadratische Gleichung sind:

x = \frac{m ^{2} - n ^{2} + m ^{4} - 6 m ^{2} n ^{2} + n ^{4}}{2 mn}, x = \frac{m ^{2} - n ^{2} - m ^{4} - 6 m ^{2} n ^{2} + n ^{4}}{2 mn}; mn \neq = 0

5 x^{2} + 7 x - 8 = 0

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

(x - 2) \cdot (x - 5) = 0

so l v e f or x, 3 x^{2} - 7 x = - 2

so l v e f or x, z = y^{2} - 1 x^{2}