AB: Gemischte Rechenaufgaben VIII

\( \begin{aligned}(2 x+1)(1-x)+\left(\frac{1}{2} x-1\right)+5 x^{2} &=(x+1)(3 x-2)+\frac{3}{2} x-1 \\ 2 x-2 x^{2}+1-x+\frac{1}{2} x-1+5 x^{2} &=3 x^{2}-2 x+3 x-2+\frac{3}{2} x-1 \\ 1,5 x+3 x^{2} &=3 x^{2}+2,5 x-3 \quad \mid-3 x \\ 1,5 x &=2,5x-3 \quad \mid-2,5 x \\ -x &=-3 \quad \mid:(-1) \\ x &=3 \end{aligned} \)

\( 2(5 x-4)(7-5 x)+(7 x-6)(2 x+2)-3 x^{2}=(9-x)(2-x)-40 x^{2}+57 \\ 2 \cdot\left(35 x-25 x^{2}-28+20 x\right)+14 x^{2}+14 x-12 x-12-3 x^{2}=(9-x)(2-x)-40 x+37 \\ 70 x-50 x^{2}-56+40 x+11 x^{2}+2 x-12=18-9 x-2 x+x^{2}-40 x^{2}+37 \\ \begin{aligned}-39 x^{2}+110 x+2 x-68 &=-39 x^{2}-11 x+55 \mid+39 x^{2} \\ 112 x-68 &=-11 x + 55 \qquad |+11x \quad |+68 \\ 123 x &=123 \qquad | :123 \\ x &=1 \end{aligned} \)

\( \begin{aligned} 4 x+0,5\left[\left(x+\frac{1}{2}\right)-(1-x)\right]+\frac{5}{2} &=\frac{1}{2}+2 x+\frac{5}{2}(x+1) \\ 4 x+0,5\left(x+\frac{1}{2}-1+x\right)+\frac{5}{2} &=\frac{1}{2}+2 x+\frac{5}{2} x+\frac{5}{2} \\ 4 x+0,5 x+\frac{1}{4}-0,5+0,5 x+\frac{5}{2} &=3+4,5 x \\ 5 x+245 &=3+4,5 x \quad|-4,5 x|-2,25 \\ 0,5 x &=0,75 \quad \mid: 0,5 \\ x &=\frac{3}{2} \end{aligned} \)

\(f(x)=-\frac{1}{2} x^{3}+\frac{1}{2} x^{2}+3 x \qquad x \in[-4 ; 4] \)

Nullstelle erraten:
\( x_{N1} = -2 \qquad f(-2) = 0 \) → Linearfaktor: \( (x + 2) \)

\( f(x) = \left(-\frac{1}{2} x^{3}+\frac{1}{2} x^{2} + 3x \right) : [x-(-2)] = \\ f(x) = \left(-\frac{1}{2} x^{3}+\frac{1}{2} x^{2} + 3x \right) : (x + 2) = -\frac{1}{2} x^{2} + \frac{3}{2 x} \\ \qquad \underline{ -\left(-\frac{1}{2} x^{3}-x^{2}\right) } \\ \qquad \qquad \frac{3}{2} x^{2} + 3x \\ \qquad \qquad \underline{ -\left(\frac{3}{2} x^{2} + 3x \right) } \\ \qquad \qquad\qquad\qquad 0 \)

\( f(x)=-\frac{1}{2} x^{2}+\frac{3}{2} x=0 \qquad | :(-\frac{1}{2}) \\ x^{2}-3 x = 0 \\ x_{1,2} = -\frac{p}{2} \pm \sqrt{\left(\frac{p^{2}}{2}\right) - q} \)

\( x_{1,2} = -\left(\frac{-3}{2}\right)+\sqrt{\left(\frac{-3}{2}\right)^{2}-0} \\ x_{1,2} = \frac{3}{2} \pm \sqrt{\frac{9}{4}} \\ x_{1,2} = \frac{3}{2} \pm \frac{3}{2} \\ x_{N2} = \frac{3}{2}+\frac{3}{2} = \frac{6}{2} = 3 \\ x_{N3} = \frac{3}{2}-\frac{3}{2} = 0 \)

Die Funktionsgleichung lautet:
\( f_{1}(x) = -\frac{1}{2}·(x+2)·(x-3)·x \)

Wertetabelle:

x	-4	-3	-2	-1	0	1	2	3	4
f1(x)	28	9	0	-2	0	3	4	0	-12

Graph:
~plot~ -0,5*(x+2)*(x-3)*x ~plot~

\( f(x)=-x^{3}-3 x^{2}+x+3 \qquad x \in[-4 ; 2] \)

Nullstelle erraten:
\( x_{N1} = -3 \qquad f(-3) = 0 \) → Linearfaktor: \( (x + 3) \)

\( f(x) = \left(-x^{3} -3x^2 + 3 + x \right) : [x-(-3)] = \\ f(x) = \left(-x^{3} -3x^2 + 3 + x \right) : (x + 3) = -x^2 + 1 \\ \qquad \underline{ -\left(-x^{3} - 3x^{2}\right) } \\ \qquad \qquad \qquad \quad 0 + x \\ \qquad \qquad \qquad \quad \quad \underline{ -(3 + x) } \\ \qquad \qquad \qquad \quad \qquad \underline{ -3 + 3 } \\ \qquad \qquad \qquad \quad \qquad \qquad \quad 0 \)

\( f(x) = -x^2 + 1 = 0 \qquad | ·(-1) \\ x^2 - 1 = 0 \qquad | +1 \\ x^2 = 1 \qquad | \sqrt{ } \\ x_{N1, N2} = \pm \sqrt{1} \)

Die Funktionsgleichung lautet:
\( f_{2}(x) = -(x+3)·(x+1)·(x-1) \)

Wertetabelle:

x	-4	-3	-2	-1	0	1	2	3	4
f2(x)	15	0	-3	0	3	0	-15	-48	-105

Graph:
~plot~ -(x+3)*(x+1)*(x-1) ~plot~

\( f(x)=\frac{1}{4} x^{4}-\frac{29}{16} x^{2}+\frac{25}{16} \qquad x \in[-3 ; 3] \)

Nullstelle erraten:
\( x_{N1} = 1 \qquad f(1) = 0 \) → Linearfaktor: \( (x - 1) \)

Mit Substition von \( x^2 = z \):
\( \frac{1}{4} z^{2}-\frac{2 y}{16} z+\frac{25}{16}=0 \quad 1: \frac{1}{4} \\ z^{2}-\frac{29}{4} z+\frac{25}{4}=0 \\ z_{1,2}=-\left(\frac{-29}{2}\right) \pm \sqrt{\frac{\left(\frac{25}{4}\right)^{2}}{27}-\frac{25}{4}} \\ = \frac{29}{8} \pm \sqrt{\frac{841}{64}-\frac{400}{64}} \\ = \frac{25}{8} \pm \sqrt{\frac{441}{64}} \\ = \frac{29}{8} \pm \frac{21}{8} \\ z_{1} = 1 \text{ und } z_{2} = \frac{50}{8} = \frac{25}{4} \)

Rücksubstitution:
\( x_{N1} = \sqrt{ z_{1} } = \sqrt{1} = 1 \\ x_{N2} = \sqrt{ \frac{25}{4} } = \frac{5}{2} \\ x_{N3} = -\sqrt{ z_{1} } = -\sqrt{1} = -1 \\ x_{N4} = -\sqrt{ \frac{25}{4} } = -\frac{5}{2} \)

Die Funktionsgleichung lautet:
\( f_{3}(x) = \frac{1}{4}(x-1)·(x+1)·(x-\frac{5}{2})·(x+\frac{5}{2}) \)

Wertetabelle:

x	-4	-3	-2	-1	0	1	2	3	4
f3(x)	36,5625	5,5	-1,6875	0	1,5625	0	-1,6875	5,5	36,5625

Graph:
~plot~ 1/4*x^4-29/16*x^2+25/16 ~plot~