Løs (x-1/3)^3-(2/5x-1)(2-x)-x(2/5x+3)=x^2(x-1)-1/3(2-x)

$$( x - \frac { 1 } { 3 } ) ^ { 3 } - ( \frac { 2 } { 5 } x - 1 ) ( 2 - x ) - x ( \frac { 2 } { 5 } x + 3 ) = x ^ { 2 } ( x - 1 ) - \frac { 1 } { 3 } ( 2 - x )$$

Løs for x

$$x=\frac{355}{648}\approx 0,547839506$$

Vejledning i at løse lineær ligning

Brug binomialsætningen \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} til at udvide \left(x-\frac{1}{3}\right)^{3}.

$$x^{3}-x^{2}+\frac{1}{3}x-\frac{1}{27}-\left(\frac{2}{5}x-1\right)\left(2-x\right)-x\left(\frac{2}{5}x+3\right)=x^{2}\left(x-1\right)-\frac{1}{3}\left(2-x\right)$$

Brug fordelingsegenskaben til at multiplicere \frac{2}{5}x-1 med 2-x, og kombiner ens led.

$$x^{3}-x^{2}+\frac{1}{3}x-\frac{1}{27}-\left(\frac{9}{5}x-\frac{2}{5}x^{2}-2\right)-x\left(\frac{2}{5}x+3\right)=x^{2}\left(x-1\right)-\frac{1}{3}\left(2-x\right)$$

For at finde det modsatte af \frac{9}{5}x-\frac{2}{5}x^{2}-2 skal du finde det modsatte af hvert led.

$$x^{3}-x^{2}+\frac{1}{3}x-\frac{1}{27}-\frac{9}{5}x+\frac{2}{5}x^{2}+2-x\left(\frac{2}{5}x+3\right)=x^{2}\left(x-1\right)-\frac{1}{3}\left(2-x\right)$$

Kombiner \frac{1}{3}x og -\frac{9}{5}x for at få -\frac{22}{15}x.

$$x^{3}-x^{2}-\frac{22}{15}x-\frac{1}{27}+\frac{2}{5}x^{2}+2-x\left(\frac{2}{5}x+3\right)=x^{2}\left(x-1\right)-\frac{1}{3}\left(2-x\right)$$

Kombiner -x^{2} og \frac{2}{5}x^{2} for at få -\frac{3}{5}x^{2}.

$$x^{3}-\frac{3}{5}x^{2}-\frac{22}{15}x-\frac{1}{27}+2-x\left(\frac{2}{5}x+3\right)=x^{2}\left(x-1\right)-\frac{1}{3}\left(2-x\right)$$

Tilføj -\frac{1}{27} og 2 for at få \frac{53}{27}.

$$x^{3}-\frac{3}{5}x^{2}-\frac{22}{15}x+\frac{53}{27}-x\left(\frac{2}{5}x+3\right)=x^{2}\left(x-1\right)-\frac{1}{3}\left(2-x\right)$$

Brug fordelingsegenskaben til at multiplicere x med \frac{2}{5}x+3.

$$x^{3}-\frac{3}{5}x^{2}-\frac{22}{15}x+\frac{53}{27}-\left(\frac{2}{5}x^{2}+3x\right)=x^{2}\left(x-1\right)-\frac{1}{3}\left(2-x\right)$$

For at finde det modsatte af \frac{2}{5}x^{2}+3x skal du finde det modsatte af hvert led.

$$x^{3}-\frac{3}{5}x^{2}-\frac{22}{15}x+\frac{53}{27}-\frac{2}{5}x^{2}-3x=x^{2}\left(x-1\right)-\frac{1}{3}\left(2-x\right)$$

Kombiner -\frac{3}{5}x^{2} og -\frac{2}{5}x^{2} for at få -x^{2}.

$$x^{3}-x^{2}-\frac{22}{15}x+\frac{53}{27}-3x=x^{2}\left(x-1\right)-\frac{1}{3}\left(2-x\right)$$

Kombiner -\frac{22}{15}x og -3x for at få -\frac{67}{15}x.

$$x^{3}-x^{2}-\frac{67}{15}x+\frac{53}{27}=x^{2}\left(x-1\right)-\frac{1}{3}\left(2-x\right)$$

Brug fordelingsegenskaben til at multiplicere x^{2} med x-1.

$$x^{3}-x^{2}-\frac{67}{15}x+\frac{53}{27}=x^{3}-x^{2}-\frac{1}{3}\left(2-x\right)$$

Brug fordelingsegenskaben til at multiplicere -\frac{1}{3} med 2-x.

$$x^{3}-x^{2}-\frac{67}{15}x+\frac{53}{27}=x^{3}-x^{2}-\frac{2}{3}+\frac{1}{3}x$$

Subtraher x^{3} fra begge sider.

$$x^{3}-x^{2}-\frac{67}{15}x+\frac{53}{27}-x^{3}=-x^{2}-\frac{2}{3}+\frac{1}{3}x$$

Kombiner x^{3} og -x^{3} for at få 0.

$$-x^{2}-\frac{67}{15}x+\frac{53}{27}=-x^{2}-\frac{2}{3}+\frac{1}{3}x$$

Tilføj x^{2} på begge sider.

$$-x^{2}-\frac{67}{15}x+\frac{53}{27}+x^{2}=-\frac{2}{3}+\frac{1}{3}x$$

Kombiner -x^{2} og x^{2} for at få 0.

$$-\frac{67}{15}x+\frac{53}{27}=-\frac{2}{3}+\frac{1}{3}x$$

Subtraher \frac{1}{3}x fra begge sider.

$$-\frac{67}{15}x+\frac{53}{27}-\frac{1}{3}x=-\frac{2}{3}$$

Kombiner -\frac{67}{15}x og -\frac{1}{3}x for at få -\frac{24}{5}x.

$$-\frac{24}{5}x+\frac{53}{27}=-\frac{2}{3}$$

Subtraher \frac{53}{27} fra begge sider.

$$-\frac{24}{5}x=-\frac{2}{3}-\frac{53}{27}$$

Subtraher \frac{53}{27} fra -\frac{2}{3} for at få -\frac{71}{27}.

$$-\frac{24}{5}x=-\frac{71}{27}$$

Multiplicer begge sider med -\frac{5}{24}, den reciprokke af -\frac{24}{5}.

$$x=-\frac{71}{27}\left(-\frac{5}{24}\right)$$

Multiplicer -\frac{71}{27} og -\frac{5}{24} for at få \frac{355}{648}.

$$x=\frac{355}{648}$$