Änderungen von Dokument Lösung Exponentialgleichungen (Substitution)
Zuletzt geändert von Kim Fujan am 2025/05/21 10:47
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... ... @@ -1,11 +1,11 @@ 1 1 (% class="abc" %) 2 -1. {{formula}} x^{2} +2x-3=0 {{/formula}}2 +1. {{formula}} x^{2}-2x-3=0 {{/formula}} 3 3 4 4 Lösung mit abc-Formel: 5 5 {{formula}}x_{1,2}=\frac{2\pm \sqrt{4+12}}{2}=\frac{2\pm 4}{2}{{/formula}} 6 6 {{formula}} x_{1}=3 \quad ; \quad x_{2}=-1 {{/formula}} 7 7 8 -1. {{formula}} e^{2x} +2e^x-3=0 {{/formula}}8 +1. {{formula}} e^{2x}-2e^x-3=0 {{/formula}} 9 9 10 10 Substitution: {{formula}} e^x=u {{/formula}} 11 11 {{formula}} u^{2}+2u-3=0 {{/formula}} ... ... @@ -13,13 +13,36 @@ 13 13 Lösung mit abc-Formel: 14 14 {{formula}}u_{1,2}=\frac{2\pm \sqrt{4+12}}{2}=\frac{2\pm 4}{2}{{/formula}} 15 15 {{formula}} u_{1}=3 \quad ; \quad u_{2}=-1 {{/formula}} 16 + 16 16 Resubstitution: 17 17 {{formula}} e^x=3 \quad \Longleftrightarrow \quad x=ln(3) {{/formula}} 18 18 {{formula}} e^x=-1 \quad \Longleftrightarrow \quad {{/formula}} keine weitere Lösung! 19 19 20 - 21 -1. {{formula}} e^x+2e^{\frac{1}{2}x}-3=0 {{/formula}} 21 +1. {{formula}} e^x-2e^{\frac{1}{2}x}-3=0 {{/formula}} 22 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 + 23 23 1. {{formula}} e^x-2-\frac{15}{e^x}}=0 {{/formula}} 24 24 25 25 1. {{formula}} 2e^{4x}=e^{2x}+3 {{/formula}} 37 +{{formula}} 2e^{4x}-e^{2x}-3=0 {{/formula}} 38 + 39 +Substitution: {{formula}} e^{2x}=u {{/formula}} 40 +{{formula}} 2u^{2}-u-3=0 {{/formula}} 41 + 42 +Lösung mit abc-Formel: 43 +{{formula}}u_{1,2}=\frac{1\pm \sqrt{1+24}}{2}=\frac{1\pm 5}{2}{{/formula}} 44 +{{formula}} u_{1}=3 \quad ; \quad u_{2}=-2 {{/formula}} 45 + 46 +Resubstitution: 47 +{{formula}} e^{\frac{1}{2}x}=3 \quad \Longleftrightarrow \quad x=2 \cdot ln(3) {{/formula}} 48 +{{formula}} e^{\frac{1}{2}x}=-2 \quad \Longleftrightarrow \quad {{/formula}} keine weitere Lösung!