Änderungen von Dokument BPE 3.4 Polynomgleichungen

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...	...	@@ -54,36 +54,48 @@
54	54
55	55	{{aufgabe id="Rückwärts lösen" afb="III" kompetenzen="K2,K5" quelle="Martina Wagner" lizenz="BY-SA"}}
56	56	(% class="abc" %)
57		-1. ((({{formula}}\square x^3+\square=0{{/formula}}
58		-{{formula}}\square x^3=\square{{/formula}} | //:2//
59		-{{formula}}x^3=\square{{/formula}}
60		-{{formula}}x=-2{{/formula}}
	57	+1. (((
	58	+{{formula}}
	59	+\begin{align*}
	60	+\square x^3+\square &= 0\\
	61	+\square x^3 &=\square\quad \left| :2\\
	62	+x^3 &= \square \\
	63	+x &= -2
	64	+\end{align*}
	65	+{{/formula}}
61	61	)))
62		-1. ((({{formula}}2x^3+\square x^2=0{{/formula}}
63		-{{formula}}\square (x-\square)=0{{/formula}} || SVNP
64		-{{formula}}x_{1,2}=\square; x_3=6{{/formula}}
	67	+1. (((
	68	+
	69	+{{formula}}
	70	+\begin{align*}
	71	+2x^3+\square x^2 &= 0 \\
	72	+\square (x-\square) &= 0 \left|\left| \text{ SVNP }
	73	+\end{align*}
	74	+{{/formula}}
	75	+
	76	+{{formula}}\Rightarrow x_{1,2}=\square; x_3=6{{/formula}}
65	65	)))
66		-1. ((({{formula}}x^4+\square x^2+\square=0{{/formula}} || Subst.: {{formula}}x^2:=\square{{/formula}}
67		-{{formula}}z^2+\square z + \square = 0{{/formula}} || SVNP
68		-{{formula fontSize="larger"}}z_{1,2}=\frac{\square\pm\sqrt{\square^2-4\cdot\square\cdot\square}}{2\cdot\square}{{/formula}}
69		-{{formula fontSize="larger"}}z_1=\frac{\square+\square}{\square}; z_2=\frac{\square-\square}{\square}{{/formula}}
70		-Resubst.: {{formula}}\square := x^2{{/formula}}
71		-{{formula}}x_{1,2}^2=36 \Rightarrow x_{1,2}=\square{{/formula}}
72		-{{formula}}x_{3,4}^2=\square \Rightarrow x_{3,4}=\pm 2{{/formula}}
73		-)))
74		-1. ((({{formula}}\begin{align*}
	78	+1. (((
	79	+
	80	+{{formula}}\begin{align*}
75	75	x^4+\square x^2+\square &= 0 & \left|\left|\text{ Subst.: } & x^2:=\square\\
76		-z^2+\square z + \square &= 0 & \left|\left|\text{ SVNP } &\\
	82	+z^2+\square z + \square &= 0 & \left|\left|\text{ SVNP } &
77	77	\end{align*}
78	78	{{/formula}}
	85	+
79	79	{{formula}}
80	80	\begin{align*}
81		-z_{1,2}=\frac{\square\pm\sqrt{\square^2-4\cdot\square\cdot\square}}{2\cdot\square}\\
82		-z_1=\frac{\square+\square}{\square}; z_2=\frac{\square-\square}{\square}\\
83		-\text{Resubst.: } \square := x^2\\
84		-x_{1,2}^2=36 \Rightarrow x_{1,2}=\square\\
85		-x_{3,4}^2=\square \Rightarrow x_{3,4}=\pm 2
	88	+\Rightarrowz_{1,2}&=\frac{\square\pm\sqrt{\square^2-4\cdot\square\cdot\square}}{2\cdot\square}\\
	89	+z_1&=\frac{\square+\square}{\square}; z_2=\frac{\square-\square}{\square}
86	86	\end{align*}
	91	+{{/formula}}
	92	+
	93	+{{formula}}
	94	+\begin{align*}
	95	+&\text{Resubst.: } \square := x^2\\
	96	+&x_{1,2}^2=36 \Rightarrow x_{1,2}=\square\\
	97	+&x_{3,4}^2=\square \Rightarrow x_{3,4}=\pm 2
	98	+\end{align*}
87	87	{{/formula}})))
88	88	{{/aufgabe}}
89	89