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Zuletzt geändert von Martina Wagner am 2025/05/20 14:31
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Martina Wagner 1.1 1
2 1. ((({{{ }}}
3
4 {{formula}}
5 \begin{align*}
6 2 e^x-2 &= 0\\
7 2 e^x &=2\quad \left|:2\\
8 e^x &=1 \\
9 x &= 0
10 \end{align*}
11 {{/formula}}
12 )))
13 1. ((({{{ }}}
14
15 {{formula}}
16 \begin{align*}
Martina Wagner 2.1 17 e^{2x}-e \cdot e^x &= 0 \\
18 e^x\cdot(e^x-e) &= 0 \left|\left| \text{ SVNP }
Martina Wagner 1.1 19 \end{align*}
20 {{/formula}}
21
Martina Wagner 2.1 22 {{formula}}\ e^x \neq 0 ~und~ e^x-e = 0{{/formula}}
23 {{formula}}\ e^x = e{{/formula}}
24 {{formula}} x =1 {{/formula}}
Martina Wagner 1.1 25 )))
26 1. ((({{{ }}}
27
28 {{formula}}
29 \begin{align*}
Martina Wagner 3.1 30 e^{2x}-4 e^x+4 &= 0 \quad \left|\left|\text{ Subst.: } e^x:=z\\
31 z^2-4 z + 4 &= 0 \quad \left|\left|\text{ Mitternachtsformel/abc-Formel } &
Martina Wagner 1.1 32 \end{align*}
33 {{/formula}}
34
35 {{formula}}
36 \begin{align*}
Martina Wagner 3.1 37 \Rightarrow z_{1,2}&=\frac{4\pm\sqrt{4^2-4\cdot 1\cdot 4}}{2\cdot 1}\\
38 z_{1,2}&=\frac{4+0}{2}=2
Martina Wagner 1.1 39 \end{align*}
40 {{/formula}}
41
42 {{formula}}
43 \begin{align*}
44 &\text{Resubst.: } z:= e^x\\
Martina Wagner 4.1 45 &e^x=2 \left|\left|\text{ ln }
46 \Rightarrow x \approx 0,693147...\\
Martina Wagner 1.1 47 \end{align*}
48 {{/formula}}
49 )))
Martina Wagner 3.1 50