AB: Gemischte Rechenaufgaben VI
I. Quadratische Terme
Klammern Sie aus folgenden Termen den größten Faktor aus.
Beispiel: \( 2 x^{2}+16 x = 2x·(x+8) \)
Probe: \( 2x·(x+8) = 2x·x + 2x·8 = 2x^2 + 16x \)
\( x^{2}-17 x \)
\( x^{2}-17 x=x(x-17) \)
\( 3 x^{2}+15 x \)
\( 3 x^{2}+15 x=3 x(x+5) \)
\( \frac{3}{2} x^{2}-\frac{7}{2} x \)
\( \frac{3}{2} x^{2}-\frac{7}{2} x=\frac{3}{2} x \left(x-\frac{7}{3}\right) \)
\( -13 x^{2}+52 x \)
\( -13 x^{2}+52 x=13 x(-x+4) \)
\( \frac{1}{2} x^{2}+4 x \)
\( \frac{1}{2} x^{2} + 4x = \frac{1}{2}x (x+8) \)
\( 5 x^{4}+25 x^{2} \)
\( 5 x^{4}+25 x^{2}=5 x^{2}\left(x^{2}+5\right) \)
II. p-q-Formel
Lösen Sie mit Hilfe der p-q-Formel die quadratischen Gleichungen.
Quadratische Gleichung: \( x^2 + px + q = 0 \)
p-q-Formel: \( x_{1,2} = -( \frac{p}{2} ) \pm \sqrt{ ( \frac{p}{2} )^{2} - q} \)
Beispielrechnung:
\(
x^{2}-5 x+6=0 \\
x_{1,2}=-\frac{(-5)}{2} \pm \sqrt{\left(\frac{(-5)}{2}\right)^{2}-6} \\
x_{1,2}=\frac{5}{2} \pm \sqrt{\frac{25}{4}-6}=\frac{5}{2} \pm \sqrt{\frac{25}{4}-\frac{24}{4}}=\frac{5}{2} \pm \sqrt{\frac{1}{4}}=\frac{5}{2} \pm \frac{1}{2} \\
x_{1}=\frac{5}{2}+\frac{1}{2}=\frac{6}{2} = 3 \\
x_{2}=\frac{5}{2}-\frac{1}{2}=\frac{4}{2} = 2
\)
\( x^2 - 3x - 4 = 0 \)
\( x^{2}-3 x-4=0 \\ x_{1,2}=-\frac{(-3)}{2} \pm \sqrt{\left(\frac{(-3)^{2}}{2}\right)^{2}-14} \\ x_{1,2}=\frac{3}{2} \pm \sqrt{\frac{9}{4}+4} \\ x_{1,2}=\frac{2}{2} \pm \sqrt{\frac{25}{4}} \\ x_{1}=\frac{3}{2} + \sqrt{\frac{25}{4}}=\frac{3}{2}+\frac{5}{2}=\frac{8}{2} = 4 \\ x_{2}=\frac{3}{2} - \sqrt{\frac{25}{4}}=\frac{3}{2} - \frac{5}{2}=-\frac{2}{2}=-1 \)
\( x^2 + 8x + 15 = 0 \)
\( x^{2}+8 x+15=0 \\ x_{1,2}=-\frac{8}{2} \pm \sqrt{\left(\frac{8}{2}\right)^{2}-15} \\ x_{1,2}=-4 \pm \sqrt{16-15} \\ x_{1} = -4 + 1 = -3 \\ x_{2} = -4 -1 = -5 \)
\( x^2 - \frac{5}{4}x + \frac{3}{8} = 0 \)
\( x^{2}-\frac{5}{4} x+\frac{3}{8}=0 \\ x_{1,2}=-\left(\frac{-\frac{5}{4}}{2}\right) \pm \sqrt{\left( \frac{ -\frac{5}{4} }{2} \right)^{2}-\frac{3}{8} } \\ x_{1,2}=\frac{5}{8} \pm \sqrt{\frac{25}{64} - \frac{3}{8} } \\ x_{1,2}=\frac{5}{8} \pm \sqrt{\frac{25}{64} - \frac{24}{64} } \\ x_{1,2}=\frac{5}{8} \pm \sqrt{ \frac{1}{64} } \\ x_{1}=\frac{5}{8}+\frac{1}{8} = \frac{6}{8} = \frac{3}{4} \\ x_{2}=\frac{5}{8}-\frac{1}{8} = \frac{4}{8} = \frac{1}{2} \)