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Zuletzt geändert von akukin am 2025/08/11 15:27
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1 (% class="abc" %)
2 1. {{formula}} x^{2}-2x-3=0 {{/formula}}
3
4 Lösung mit abc-Formel:
5 {{formula}}x_{1,2}=\frac{2\pm \sqrt{4+12}}{2}=\frac{2\pm 4}{2}{{/formula}}
6 {{formula}} x_{1}=3 \quad ; \quad x_{2}=-1 {{/formula}}
7
8 1. {{formula}} e^{2x}-2e^x-3=0 {{/formula}}
9
10 Substitution: {{formula}} e^x=u {{/formula}}
11 {{formula}} u^{2}+2u-3=0 {{/formula}}
12
13 Lösung mit abc-Formel:
14 {{formula}}u_{1,2}=\frac{2\pm \sqrt{4+12}}{2}=\frac{2\pm 4}{2}{{/formula}}
15 {{formula}} u_{1}=3 \quad ; \quad u_{2}=-1 {{/formula}}
16
17 Resubstitution:
18 {{formula}} e^x=3 \quad \Longleftrightarrow \quad x=ln(3) {{/formula}}
19 {{formula}} e^x=-1 \quad \Longleftrightarrow \quad {{/formula}} keine weitere Lösung!
20
21 1. {{formula}} e^x-2e^{\frac{1}{2}x}-3=0 {{/formula}}
22
23 Substitution: {{formula}} e^{\frac{1}{2}x}=u {{/formula}}
24 {{formula}} u^{2}+2u-3=0 {{/formula}}
25
26 Lösung mit abc-Formel:
27 {{formula}}u_{1,2}=\frac{2\pm \sqrt{4+12}}{2}=\frac{2\pm 4}{2}{{/formula}}
28 {{formula}} u_{1}=3 \quad ; \quad u_{2}=-1 {{/formula}}
29
30 Resubstitution:
31 {{formula}} e^{\frac{1}{2}x}=3 \quad \Longleftrightarrow \quad x=2 \cdot ln(3) {{/formula}}
32 {{formula}} e^{\frac{1}{2}x}=-1 \quad \Longleftrightarrow \quad {{/formula}} keine weitere Lösung!
33
34 1. {{formula}} e^x-2-\frac{8}{e^x}=0 {{/formula}}
35 {{formula}} e^{-x} \cdot (e^{2x}-2e^x-8)=0 {{/formula}}
36
37 Substitution: {{formula}} e^{x}=u {{/formula}}
38 {{formula}} u^{2}-2u-8=0 {{/formula}}
39
40 Lösung mit abc-Formel:
41 {{formula}}u_{1,2}=\frac{2\pm \sqrt{4+32}}{2}=\frac{2\pm 6}{2}{{/formula}}
42 {{formula}} u_{1}=4 \quad ; \quad u_{2}=-2 {{/formula}}
43
44 Resubstitution:
45 {{formula}} e^{x}=4 \quad \Longleftrightarrow \quad x= ln(4) {{/formula}}
46 {{formula}} e^{x}=-2 \quad \Longleftrightarrow \quad {{/formula}} keine weitere Lösung!
47
48 1. {{formula}} 2e^{4x}=e^{2x}+3 {{/formula}}
49 {{formula}} 2e^{4x}-e^{2x}-3=0 {{/formula}}
50
51 Substitution: {{formula}} e^{2x}=u {{/formula}}
52 {{formula}} 2u^{2}-u-3=0 {{/formula}}
53
54 Lösung mit abc-Formel:
55 {{formula}}u_{1,2}=\frac{1\pm \sqrt{1+24}}{2}=\frac{1\pm 5}{2}{{/formula}}
56 {{formula}} u_{1}=3 \quad ; \quad u_{2}=-2 {{/formula}}
57
58 Resubstitution:
59 {{formula}} e^{2x}=3 \quad \Longleftrightarrow \quad x=\frac{1}{2} \cdot ln(3) {{/formula}}
60 {{formula}} e^{2x}=-2 \quad \Longleftrightarrow \quad {{/formula}} keine weitere Lösung!