(x-7)^2+(mx+c-9)^2=9

Lösung

(x - 7)^{2} + (m x + c - 9)^{2} = 9

x = \frac{- m c + 9 m + 7 + - 40 m ^{2} + 126 m - 14 m c + 18 c - c ^{2} - 72}{m ^{2} + 1}, x = \frac{- m c + 9 m + 7 - - 40 m ^{2} + 126 m - 14 m c + 18 c - c ^{2} - 72}{m ^{2} + 1}; m \neq = i, m \neq = - i

Schritte zur Lösung

(x - 7)^{2} + (m x + c - 9)^{2} = 9

Schreibe

(x - 7)^{2} + (m x + c - 9)^{2}

um:

m^{2} x^{2} + x^{2} + 2 m c x - 18 m x - 14 x + c^{2} + 130 - 18 c

m^{2} x^{2} + x^{2} + 2 m c x - 18 m x - 14 x + c^{2} + 130 - 18 c = 9

Verschiebe

9

auf die linke Seite

m^{2} x^{2} + x^{2} + 2 m c x - 18 m x - 14 x + c^{2} + 121 - 18 c = 0

Schreibe in der Standard Form

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

(m^{2} + 1) x^{2} + (2 m c - 18 m - 14) x + c^{2} + 121 - 18 c = 0

Löse mit der quadratischen Formel

x_{1, 2} = \frac{- ( 2 m c - 18 m - 14 ) \pm ( 2 m c - 18 m - 14 ) ^{2} - 4 ( m ^{2} + 1 ) ( c ^{2} + 121 - 18 c )}{2 ( m ^{2} + 1 )}

Vereinfache

(2 m c - 18 m - 14)^{2} - 4 (m^{2} + 1) (c^{2} + 121 - 18 c) : 2 - 40 m^{2} + 126 m - 14 m c + 18 c - c^{2} - 72

x_{1, 2} = \frac{- ( 2 m c - 18 m - 14 ) \pm 2 - 40 m ^{2} + 126 m - 14 m c + 18 c - c ^{2} - 72}{2 ( m ^{2} + 1 )}; m \neq = i, m \neq = - i

Trenne die Lösungen

x_{1} = \frac{- ( 2 m c - 18 m - 14 ) + 2 - 40 m ^{2} + 126 m - 14 m c + 18 c - c ^{2} - 72}{2 ( m ^{2} + 1 )}, x_{2} = \frac{- ( 2 m c - 18 m - 14 ) - 2 - 40 m ^{2} + 126 m - 14 m c + 18 c - c ^{2} - 72}{2 ( m ^{2} + 1 )}

x = \frac{- ( 2 m c - 18 m - 14 ) + 2 - 40 m ^{2} + 126 m - 14 m c + 18 c - c ^{2} - 72}{2 ( m ^{2} + 1 )} : \frac{- m c + 9 m + 7 + - 40 m ^{2} + 126 m - 14 m c + 18 c - c ^{2} - 72}{m ^{2} + 1}

x = \frac{- ( 2 m c - 18 m - 14 ) - 2 - 40 m ^{2} + 126 m - 14 m c + 18 c - c ^{2} - 72}{2 ( m ^{2} + 1 )} : \frac{- m c + 9 m + 7 - - 40 m ^{2} + 126 m - 14 m c + 18 c - c ^{2} - 72}{m ^{2} + 1}

Die Lösungen für die quadratische Gleichung sind:

x = \frac{- m c + 9 m + 7 + - 40 m ^{2} + 126 m - 14 m c + 18 c - c ^{2} - 72}{m ^{2} + 1}, x = \frac{- m c + 9 m + 7 - - 40 m ^{2} + 126 m - 14 m c + 18 c - c ^{2} - 72}{m ^{2} + 1}; m \neq = i, m \neq = - i

(x - 3)^{2} = - 6 x + 12

(- \frac{1}{4})^{2} + x^{2} = 1

x^{2} - 12 x + 2 = - 6

0 = - 2 x^{2} + 16 x - 24

so l v e f or x, x^{2} - m x + 36 = 0