solvefor x,4x^2+9(mx-1)^2=5

Lösung

x, 4 x^{2} + 9 (m x - 1)^{2} = 5

x = \frac{9 m + 45 m ^{2} - 16}{4 + 9 m ^{2}}, x = \frac{9 m - 45 m ^{2} - 16}{4 + 9 m ^{2}}; m \neq = \frac{2}{3} i, m \neq = - \frac{2}{3} i

Schritte zur Lösung

4 x^{2} + 9 (m x - 1)^{2} = 5

Schreibe

4 x^{2} + 9 (m x - 1)^{2}

um:

4 x^{2} + 9 m^{2} x^{2} - 18 m x + 9

4 x^{2} + 9 m^{2} x^{2} - 18 m x + 9 = 5

Verschiebe

5

auf die linke Seite

4 x^{2} + 9 m^{2} x^{2} - 18 m x + 4 = 0

Schreibe in der Standard Form

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

(4 + 9 m^{2}) x^{2} - 18 m x + 4 = 0

Löse mit der quadratischen Formel

x_{1, 2} = \frac{- ( - 18 m ) \pm ( - 18 m ) ^{2} - 4 ( 4 + 9 m ^{2} ) \cdot 4}{2 ( 4 + 9 m ^{2} )}

Vereinfache

(- 18 m)^{2} - 4 (4 + 9 m^{2}) \cdot 4 : 2 45 m^{2} - 16

x_{1, 2} = \frac{- ( - 18 m ) \pm 2 45 m ^{2} - 16}{2 ( 4 + 9 m ^{2} )}; m \neq = \frac{2}{3} i, m \neq = - \frac{2}{3} i

Trenne die Lösungen

x_{1} = \frac{- ( - 18 m ) + 2 45 m ^{2} - 16}{2 ( 4 + 9 m ^{2} )}, x_{2} = \frac{- ( - 18 m ) - 2 45 m ^{2} - 16}{2 ( 4 + 9 m ^{2} )}

x = \frac{- ( - 18 m ) + 2 45 m ^{2} - 16}{2 ( 4 + 9 m ^{2} )} : \frac{9 m + 45 m ^{2} - 16}{4 + 9 m ^{2}}

x = \frac{- ( - 18 m ) - 2 45 m ^{2} - 16}{2 ( 4 + 9 m ^{2} )} : \frac{9 m - 45 m ^{2} - 16}{4 + 9 m ^{2}}

Die Lösungen für die quadratische Gleichung sind:

x = \frac{9 m + 45 m ^{2} - 16}{4 + 9 m ^{2}}, x = \frac{9 m - 45 m ^{2} - 16}{4 + 9 m ^{2}}; m \neq = \frac{2}{3} i, m \neq = - \frac{2}{3} i

2 5^{2} = 7^{2} + b^{2}

- 26 = - x^{2} - 1

9 (x^{2} + 1) = 34

\frac{x ^{2}}{2} = 7 x

x^{2} - 3 x + (- \frac{9}{4}) = 12