Wiki-Quellcode von Lösung Rückwärts lösen
Verstecke letzte Bearbeiter
| author | version | line-number | content |
|---|---|---|---|
 |
3.1 | 1 | a) Aus {{formula}}x=-2{{/formula}} ergibt sich durch Potenzieren mit {{formula}}3{{/formula}}: {{formula}}x^3=-8{{/formula}} |
| 2 | |||
| 3 | Multiplizieren mit {{formula}}2{{/formula}} ergibt {{formula}}2x^3=-16{{/formula}} | ||
| 4 | |||
| 5 | Addieren von {{formula}}16{{/formula}} auf beiden Seiten ergibt {{formula}}2x^3+16=0{{/formula}} | ||
| 6 | |||
| |
6.1 | 7 | |
| |
3.1 | 8 | Insgesamt folgt also: |
| 9 | |||
| 10 | {{formula}} | ||
| 11 | \begin{align} | ||
| 12 | 2x^3+16&=0 \\ | ||
| 13 | 2x^3&=-16 \quad \mid :2 \\ | ||
| 14 | x^3&=-8 \\ | ||
| 15 | x&=-2 | ||
| 16 | \end{align} | ||
| 17 | {{/formula}} | ||
| 18 | |||
| |
6.1 | 19 | |
| |
3.1 | 20 | b) Die Gleichung hat die Lösungen {{formula}}x_{1,2}=0{{/formula}} und {{formula}}x_3=6{{/formula}}. In der Produktform/Nullstellenform ergibt sich: |
| 21 | |||
| 22 | {{formula}} | ||
| 23 | \begin{align} | ||
| |
6.1 | 24 | 2x^2(x-6)=0 \\ |
| |
3.2 | 25 | 2x^3-12x^2=0 |
| |
3.1 | 26 | \end{align} |
| 27 | {{/formula}} | ||
| 28 | |||
| |
6.1 | 29 | |
| 30 | Insgesamt ergibt sich: | ||
| 31 | |||
| 32 | {{formula}} | ||
| 33 | \begin{align*} | ||
| 34 | 2x^3+(-12)x^2 &= 0 \\ | ||
| 35 | 2x^2 (x-6) &= 0 \left|\left| \text{ SVNP } | ||
| 36 | \end{align*} | ||
| 37 | {{/formula}} | ||
| 38 | |||
| 39 | {{formula}}\Rightarrow x_{1,2}=0; x_3=6{{/formula}} | ||
| 40 | |||
| 41 | |||
| |
3.1 | 42 | c) {{formula}}(\pm 2)^2=4{{/formula}} |
| |
6.1 | 43 | Damit ergeben sich die beiden Lösungen {{formula}}z_1=36{{/formula}} und {{formula}}z_2=4{{/formula}}. In der Produktform/Nullstellenform ergibt sich {{formula}}(z-36)(z-4)=0{{/formula}}. |
| 44 | Ausmultiplizieren führt auf | ||
| |
3.1 | 45 | |
| 46 | {{formula}} | ||
| 47 | \begin{align} | ||
| 48 | z^2-4z-36z+144=0 \\ | ||
| |
6.1 | 49 | z^2-40z+144=0 |
| |
3.1 | 50 | \end{align} |
| 51 | {{/formula}} | ||
| 52 | |||
| |
3.2 | 53 | |
| 54 | |||
| |
5.1 | 55 | Insgesamt ergibt sich |
| |
3.1 | 56 | |
| 57 | {{formula}}\begin{align*} | ||
| |
6.1 | 58 | x^4-40x^2+144 &= 0 \quad \left|\left|\text{ Subst.: } x^2:=\square\\ |
| 59 | z^2-40z+144 &= 0 \quad \left|\left|\text{ Mitternachtsformel/abc-Formel } & | ||
| |
3.1 | 60 | \end{align*} |
| 61 | {{/formula}} | ||
| 62 | |||
| 63 | {{formula}} | ||
| 64 | \begin{align*} | ||
| |
4.1 | 65 | \Rightarrow z_{1,2}&=\frac{40\pm\sqrt{(-40)^2-4\cdot 1\cdot 144}}{2\cdot 1}\\ |
| 66 | z_1&=\frac{40+32}{2}=36; z_2=\frac{40-32}{2}=4 | ||
| |
3.1 | 67 | \end{align*} |
| 68 | {{/formula}} | ||
| 69 | |||
| 70 | {{formula}} | ||
| 71 | \begin{align*} | ||
| |
4.1 | 72 | &\text{Resubst.: } z := x^2\\ |
| 73 | &x_{1,2}^2=36 \Rightarrow x_{1,2}=\pm 6\\ | ||
| 74 | &x_{3,4}^2=4 \Rightarrow x_{3,4}=\pm 2 | ||
| |
3.1 | 75 | \end{align*} |
| 76 | {{/formula}} | ||
| 77 |