Réitigh x^3-1/(x-1)^2-x^3+1/(x+1)^2=6

$$\frac { x ^ { 3 } - 1 } { ( x - 1 ) ^ { 2 } } - \frac { x ^ { 3 } + 1 } { ( x + 1 ) ^ { 2 } } = 6$$

Réitigh do x.

$$x=2$$

$$x=-2$$

Céimeanna ag Baint Úsáid as an bhFoirmle Chearnach

Ní féidir leis an athróg x a bheith comhionann le haon cheann de na luachanna -1,1 toisc nach bhfuil an roinnt faoi nialas sainithe. Iolraigh an dá thaobh den chothromóid faoi \left(x-1\right)^{2}\left(x+1\right)^{2}, an comhiolraí is lú de \left(x-1\right)^{2},\left(x+1\right)^{2}.

$$\left(x+1\right)^{2}\left(x^{3}-1\right)-\left(x-1\right)^{2}\left(x^{3}+1\right)=6\left(x-1\right)^{2}\left(x+1\right)^{2}$$

Úsáid an teoirim dhéthéarmach \left(a+b\right)^{2}=a^{2}+2ab+b^{2} chun \left(x+1\right)^{2} a leathnú.

$$\left(x^{2}+2x+1\right)\left(x^{3}-1\right)-\left(x-1\right)^{2}\left(x^{3}+1\right)=6\left(x-1\right)^{2}\left(x+1\right)^{2}$$

Úsáid an t-airí dáileach chun x^{2}+2x+1 a mhéadú faoi x^{3}-1.

$$x^{5}-x^{2}+2x^{4}-2x+x^{3}-1-\left(x-1\right)^{2}\left(x^{3}+1\right)=6\left(x-1\right)^{2}\left(x+1\right)^{2}$$

Úsáid an teoirim dhéthéarmach \left(a-b\right)^{2}=a^{2}-2ab+b^{2} chun \left(x-1\right)^{2} a leathnú.

$$x^{5}-x^{2}+2x^{4}-2x+x^{3}-1-\left(x^{2}-2x+1\right)\left(x^{3}+1\right)=6\left(x-1\right)^{2}\left(x+1\right)^{2}$$

Úsáid an t-airí dáileach chun x^{2}-2x+1 a mhéadú faoi x^{3}+1.

$$x^{5}-x^{2}+2x^{4}-2x+x^{3}-1-\left(x^{5}+x^{2}-2x^{4}-2x+x^{3}+1\right)=6\left(x-1\right)^{2}\left(x+1\right)^{2}$$

Chun an mhalairt ar x^{5}+x^{2}-2x^{4}-2x+x^{3}+1 a aimsiú, aimsigh an mhalairt ar gach téarma.

$$x^{5}-x^{2}+2x^{4}-2x+x^{3}-1-x^{5}-x^{2}+2x^{4}+2x-x^{3}-1=6\left(x-1\right)^{2}\left(x+1\right)^{2}$$

Comhcheangail x^{5} agus -x^{5} chun 0 a fháil.

$$-x^{2}+2x^{4}-2x+x^{3}-1-x^{2}+2x^{4}+2x-x^{3}-1=6\left(x-1\right)^{2}\left(x+1\right)^{2}$$

Comhcheangail -x^{2} agus -x^{2} chun -2x^{2} a fháil.

$$-2x^{2}+2x^{4}-2x+x^{3}-1+2x^{4}+2x-x^{3}-1=6\left(x-1\right)^{2}\left(x+1\right)^{2}$$

Comhcheangail 2x^{4} agus 2x^{4} chun 4x^{4} a fháil.

$$-2x^{2}+4x^{4}-2x+x^{3}-1+2x-x^{3}-1=6\left(x-1\right)^{2}\left(x+1\right)^{2}$$

Comhcheangail -2x agus 2x chun 0 a fháil.

$$-2x^{2}+4x^{4}+x^{3}-1-x^{3}-1=6\left(x-1\right)^{2}\left(x+1\right)^{2}$$

Comhcheangail x^{3} agus -x^{3} chun 0 a fháil.

$$-2x^{2}+4x^{4}-1-1=6\left(x-1\right)^{2}\left(x+1\right)^{2}$$

Dealaigh 1 ó -1 chun -2 a fháil.

$$-2x^{2}+4x^{4}-2=6\left(x-1\right)^{2}\left(x+1\right)^{2}$$

Úsáid an teoirim dhéthéarmach \left(a-b\right)^{2}=a^{2}-2ab+b^{2} chun \left(x-1\right)^{2} a leathnú.

$$-2x^{2}+4x^{4}-2=6\left(x^{2}-2x+1\right)\left(x+1\right)^{2}$$

Úsáid an teoirim dhéthéarmach \left(a+b\right)^{2}=a^{2}+2ab+b^{2} chun \left(x+1\right)^{2} a leathnú.

$$-2x^{2}+4x^{4}-2=6\left(x^{2}-2x+1\right)\left(x^{2}+2x+1\right)$$

Úsáid an t-airí dáileach chun 6 a mhéadú faoi x^{2}-2x+1.

$$-2x^{2}+4x^{4}-2=\left(6x^{2}-12x+6\right)\left(x^{2}+2x+1\right)$$

Úsáid an t-airí dáileach chun 6x^{2}-12x+6 a mhéadú faoi x^{2}+2x+1 agus chun téarmaí comhchosúla a chumasc.

$$-2x^{2}+4x^{4}-2=6x^{4}-12x^{2}+6$$

Bain 6x^{4} ón dá thaobh.

$$-2x^{2}+4x^{4}-2-6x^{4}=-12x^{2}+6$$

Comhcheangail 4x^{4} agus -6x^{4} chun -2x^{4} a fháil.

$$-2x^{2}-2x^{4}-2=-12x^{2}+6$$

Cuir 12x^{2} leis an dá thaobh.

$$-2x^{2}-2x^{4}-2+12x^{2}=6$$

Comhcheangail -2x^{2} agus 12x^{2} chun 10x^{2} a fháil.

$$10x^{2}-2x^{4}-2=6$$

Bain 6 ón dá thaobh.

$$10x^{2}-2x^{4}-2-6=0$$

Dealaigh 6 ó -2 chun -8 a fháil.

$$10x^{2}-2x^{4}-8=0$$

Cuir t in ionad x^{2}.

$$-2t^{2}+10t-8=0$$

Is féidir gach cothromóid i bhfoirm ax^{2}+bx+c=0 a réiteach ach an fhoirmle chearnach seo a úsáid: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Cuir -2 in ionad a, 10 in ionad b agus -8 in ionad c san fhoirmle chearnach.

$$t=\frac{-10±\sqrt{10^{2}-4\left(-2\right)\left(-8\right)}}{-2\times 2}$$

Déan áirimh.

$$t=\frac{-10±6}{-4}$$

Réitigh an chothromóid t=\frac{-10±6}{-4} nuair is ionann ± agus luach deimhneach agus ± agus luach diúltach.

$$t=1 t=4$$

Más x=t^{2}, is féidir teacht ar na réitigh ach x=±\sqrt{t} a mheas i gcomhair gach t.

$$x=1 x=-1 x=2 x=-2$$

Ní féidir leis an athróg x a bheith comhionann le haon cheann de na luachanna 1,-1.

$$x=-2 x=2$$