solvefor x,(x-a)^2+((m*x+n)-b)^2=r^2

Lösung

löse nach

x, (x - a)^{2} + ((m \cdot x + n) - b)^{2} = r^{2}

x = \frac{a - mn + mb + r ^{2} + m ^{2} r ^{2} + 2 amb + 2 nb - a ^{2} m ^{2} - 2 amn - n ^{2} - b ^{2}}{1 + m ^{2}}, x = \frac{a - mn + mb - r ^{2} + m ^{2} r ^{2} + 2 amb + 2 nb - a ^{2} m ^{2} - 2 amn - n ^{2} - b ^{2}}{1 + m ^{2}}; m \neq = i, m \neq = - i

Schritte zur Lösung

(x - a)^{2} + ((m x + n) - b)^{2} = r^{2}

Schreibe

(x - a)^{2} + ((m x + n) - b)^{2}

um:

x^{2} - 2 a x + a^{2} + m^{2} x^{2} + 2 mn x - 2 mb x + n^{2} - 2 nb + b^{2}

x^{2} - 2 a x + a^{2} + m^{2} x^{2} + 2 mn x - 2 mb x + n^{2} - 2 nb + b^{2} = r^{2}

Verschiebe

r^{2}

auf die linke Seite

x^{2} - 2 a x + a^{2} + m^{2} x^{2} + 2 mn x - 2 mb x + n^{2} - 2 nb + b^{2} - r^{2} = 0

Schreibe in der Standard Form

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

(1 + m^{2}) x^{2} + (- 2 a + 2 mn - 2 mb) x + a^{2} + n^{2} - 2 nb + b^{2} - r^{2} = 0

Löse mit der quadratischen Formel

x_{1, 2} = \frac{- ( - 2 a + 2 mn - 2 mb ) \pm ( - 2 a + 2 mn - 2 mb ) ^{2} - 4 ( 1 + m ^{2} ) ( a ^{2} + n ^{2} - 2 nb + b ^{2} - r ^{2} )}{2 ( 1 + m ^{2} )}

Vereinfache

(- 2 a + 2 mn - 2 mb)^{2} - 4 (1 + m^{2}) (a^{2} + n^{2} - 2 nb + b^{2} - r^{2}) : 2 m^{2} r^{2} + r^{2} - a^{2} m^{2} - n^{2} - 2 amn - b^{2} + 2 amb + 2 nb

x_{1, 2} = \frac{- ( - 2 a + 2 mn - 2 mb ) \pm 2 m ^{2} r ^{2} + r ^{2} - a ^{2} m ^{2} - n ^{2} - 2 amn - b ^{2} + 2 amb + 2 nb}{2 ( 1 + m ^{2} )}; m \neq = i, m \neq = - i

Trenne die Lösungen

x_{1} = \frac{- ( - 2 a + 2 mn - 2 mb ) + 2 m ^{2} r ^{2} + r ^{2} - a ^{2} m ^{2} - n ^{2} - 2 amn - b ^{2} + 2 amb + 2 nb}{2 ( 1 + m ^{2} )}, x_{2} = \frac{- ( - 2 a + 2 mn - 2 mb ) - 2 m ^{2} r ^{2} + r ^{2} - a ^{2} m ^{2} - n ^{2} - 2 amn - b ^{2} + 2 amb + 2 nb}{2 ( 1 + m ^{2} )}

x = \frac{- ( - 2 a + 2 mn - 2 mb ) + 2 m ^{2} r ^{2} + r ^{2} - a ^{2} m ^{2} - n ^{2} - 2 amn - b ^{2} + 2 amb + 2 nb}{2 ( 1 + m ^{2} )} : \frac{a - mn + mb + r ^{2} + m ^{2} r ^{2} + 2 amb + 2 nb - a ^{2} m ^{2} - 2 amn - n ^{2} - b ^{2}}{1 + m ^{2}}

x = \frac{- ( - 2 a + 2 mn - 2 mb ) - 2 m ^{2} r ^{2} + r ^{2} - a ^{2} m ^{2} - n ^{2} - 2 amn - b ^{2} + 2 amb + 2 nb}{2 ( 1 + m ^{2} )} : \frac{a - mn + mb - r ^{2} + m ^{2} r ^{2} + 2 amb + 2 nb - a ^{2} m ^{2} - 2 amn - n ^{2} - b ^{2}}{1 + m ^{2}}

Die Lösungen für die quadratische Gleichung sind:

x = \frac{a - mn + mb + r ^{2} + m ^{2} r ^{2} + 2 amb + 2 nb - a ^{2} m ^{2} - 2 amn - n ^{2} - b ^{2}}{1 + m ^{2}}, x = \frac{a - mn + mb - r ^{2} + m ^{2} r ^{2} + 2 amb + 2 nb - a ^{2} m ^{2} - 2 amn - n ^{2} - b ^{2}}{1 + m ^{2}}; m \neq = i, m \neq = - i

1.16 x^{2} + 0.64 x - 0.36 = 0

1 = 2 x^{2} + 5 x

\frac{( x \cdot ( \frac{x}{4} + 1 ) )}{2} = 60

q u a d r a t i c f or m u l a \frac{x ^{2}}{4} + 3 = 2 (x - 2)

2 a^{2} - 7 a + 3 = 0