Wiki-Quellcode von Lösung Exponentialgleichungen (Substitution)
Verstecke letzte Bearbeiter
author | version | line-number | content |
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2.1 | 1 | (% class="abc" %) |
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6.1 | 2 | 1. {{formula}} x^{2}-2x-3=0 {{/formula}} |
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4.1 | 3 | |
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2.1 | 4 | Lösung mit abc-Formel: |
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3.1 | 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}} | ||
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4.1 | 7 | |
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6.1 | 8 | 1. {{formula}} e^{2x}-2e^x-3=0 {{/formula}} |
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2.1 | 9 | |
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4.1 | 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}} | ||
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5.1 | 16 | |
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4.1 | 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 | |||
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6.1 | 21 | 1. {{formula}} e^x-2e^{\frac{1}{2}x}-3=0 {{/formula}} |
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2.1 | 22 | |
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5.1 | 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! | ||
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6.1 | 33 | |
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8.1 | 34 | 1. {{formula}} e^x-2-\frac{8}{e^x}}=0 {{/formula}} |
35 | {{formula}} e^{-x} \cdot (e^{2x}-2e^x-8)=0 {{/formula}} | ||
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2.1 | 36 | |
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8.1 | 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: | ||
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8.2 | 45 | {{formula}} e^{x}=4 \quad \Longleftrightarrow \quad x= ln(4) {{/formula}} |
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8.1 | 46 | {{formula}} e^{x}=-2 \quad \Longleftrightarrow \quad {{/formula}} keine weitere Lösung! |
47 | |||
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1.1 | 48 | 1. {{formula}} 2e^{4x}=e^{2x}+3 {{/formula}} |
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6.1 | 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: | ||
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10.1 | 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! |