Issolvi {c}{(1-2/x+1)divx^2-2x+1/x+1=y}{x=sqrt{2}+1}

$$\left. \begin{array} { c } { ( 1 - \frac { 2 } { x + 1 } ) \div \frac { x ^ { 2 } - 2 x + 1 } { x + 1 } = y } \\ { x = \sqrt { 2 } + 1 } \end{array} \right.$$

Solvi għal x, y

$$x=\sqrt{2}+1\approx 2.414213562$$

$$y=\frac{\sqrt{2}}{2}\approx 0.707106781$$

Passi tas-Soluzzjonijiet

Ikkunsidra l-ewwel ekwazzjoni. Inserixxi l-valuri magħrufa tal-varjabbli fl-ekwazzjoni.

$$\frac{1-\frac{2}{\sqrt{2}+1+1}}{\frac{\left(\sqrt{2}+1\right)^{2}-2\left(\sqrt{2}+1\right)+1}{\sqrt{2}+1+1}}=y$$

Żid 1 u 1 biex tikseb 2.

$$\frac{1-\frac{2}{\sqrt{2}+2}}{\frac{\left(\sqrt{2}+1\right)^{2}-2\left(\sqrt{2}+1\right)+1}{\sqrt{2}+1+1}}=y$$

Irrazzjonalizza d-denominatur tal-\frac{2}{\sqrt{2}+2} billi timmultiplika in-numeratur u d-denominatur mill-\sqrt{2}-2.

$$\frac{1-\frac{2\left(\sqrt{2}-2\right)}{\left(\sqrt{2}+2\right)\left(\sqrt{2}-2\right)}}{\frac{\left(\sqrt{2}+1\right)^{2}-2\left(\sqrt{2}+1\right)+1}{\sqrt{2}+1+1}}=y$$

Ikkunsidra li \left(\sqrt{2}+2\right)\left(\sqrt{2}-2\right). Il-multiplikazzjoni tista' tiġi ttrasformata fid-differenza tal-kwadrati li jużaw ir-regola: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{1-\frac{2\left(\sqrt{2}-2\right)}{\left(\sqrt{2}\right)^{2}-2^{2}}}{\frac{\left(\sqrt{2}+1\right)^{2}-2\left(\sqrt{2}+1\right)+1}{\sqrt{2}+1+1}}=y$$

Ikkwadra \sqrt{2}. Ikkwadra 2.

$$\frac{1-\frac{2\left(\sqrt{2}-2\right)}{2-4}}{\frac{\left(\sqrt{2}+1\right)^{2}-2\left(\sqrt{2}+1\right)+1}{\sqrt{2}+1+1}}=y$$

Naqqas 4 minn 2 biex tikseb -2.

$$\frac{1-\frac{2\left(\sqrt{2}-2\right)}{-2}}{\frac{\left(\sqrt{2}+1\right)^{2}-2\left(\sqrt{2}+1\right)+1}{\sqrt{2}+1+1}}=y$$

Annulla -2 u -2.

$$\frac{1-\left(-\left(\sqrt{2}-2\right)\right)}{\frac{\left(\sqrt{2}+1\right)^{2}-2\left(\sqrt{2}+1\right)+1}{\sqrt{2}+1+1}}=y$$

L-oppost ta' -\left(\sqrt{2}-2\right) huwa \sqrt{2}-2.

$$\frac{1+\sqrt{2}-2}{\frac{\left(\sqrt{2}+1\right)^{2}-2\left(\sqrt{2}+1\right)+1}{\sqrt{2}+1+1}}=y$$

Naqqas 2 minn 1 biex tikseb -1.

$$\frac{-1+\sqrt{2}}{\frac{\left(\sqrt{2}+1\right)^{2}-2\left(\sqrt{2}+1\right)+1}{\sqrt{2}+1+1}}=y$$

Uża teorema binomjali \left(a+b\right)^{2}=a^{2}+2ab+b^{2} biex tespandi \left(\sqrt{2}+1\right)^{2}.

$$\frac{-1+\sqrt{2}}{\frac{\left(\sqrt{2}\right)^{2}+2\sqrt{2}+1-2\left(\sqrt{2}+1\right)+1}{\sqrt{2}+1+1}}=y$$

Il-kwadrat ta' \sqrt{2} huwa 2.

$$\frac{-1+\sqrt{2}}{\frac{2+2\sqrt{2}+1-2\left(\sqrt{2}+1\right)+1}{\sqrt{2}+1+1}}=y$$

Żid 2 u 1 biex tikseb 3.

$$\frac{-1+\sqrt{2}}{\frac{3+2\sqrt{2}-2\left(\sqrt{2}+1\right)+1}{\sqrt{2}+1+1}}=y$$

Żid 3 u 1 biex tikseb 4.

$$\frac{-1+\sqrt{2}}{\frac{4+2\sqrt{2}-2\left(\sqrt{2}+1\right)}{\sqrt{2}+1+1}}=y$$

Żid 1 u 1 biex tikseb 2.

$$\frac{-1+\sqrt{2}}{\frac{4+2\sqrt{2}-2\left(\sqrt{2}+1\right)}{\sqrt{2}+2}}=y$$

Irrazzjonalizza d-denominatur tal-\frac{4+2\sqrt{2}-2\left(\sqrt{2}+1\right)}{\sqrt{2}+2} billi timmultiplika in-numeratur u d-denominatur mill-\sqrt{2}-2.

$$\frac{-1+\sqrt{2}}{\frac{\left(4+2\sqrt{2}-2\left(\sqrt{2}+1\right)\right)\left(\sqrt{2}-2\right)}{\left(\sqrt{2}+2\right)\left(\sqrt{2}-2\right)}}=y$$

Ikkunsidra li \left(\sqrt{2}+2\right)\left(\sqrt{2}-2\right). Il-multiplikazzjoni tista' tiġi ttrasformata fid-differenza tal-kwadrati li jużaw ir-regola: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{-1+\sqrt{2}}{\frac{\left(4+2\sqrt{2}-2\left(\sqrt{2}+1\right)\right)\left(\sqrt{2}-2\right)}{\left(\sqrt{2}\right)^{2}-2^{2}}}=y$$

Ikkwadra \sqrt{2}. Ikkwadra 2.

$$\frac{-1+\sqrt{2}}{\frac{\left(4+2\sqrt{2}-2\left(\sqrt{2}+1\right)\right)\left(\sqrt{2}-2\right)}{2-4}}=y$$

Naqqas 4 minn 2 biex tikseb -2.

$$\frac{-1+\sqrt{2}}{\frac{\left(4+2\sqrt{2}-2\left(\sqrt{2}+1\right)\right)\left(\sqrt{2}-2\right)}{-2}}=y$$

Iddividi -1+\sqrt{2} b'\frac{\left(4+2\sqrt{2}-2\left(\sqrt{2}+1\right)\right)\left(\sqrt{2}-2\right)}{-2} billi timmultiplika -1+\sqrt{2} bir-reċiproku ta' \frac{\left(4+2\sqrt{2}-2\left(\sqrt{2}+1\right)\right)\left(\sqrt{2}-2\right)}{-2}.

$$\frac{\left(-1+\sqrt{2}\right)\left(-2\right)}{\left(4+2\sqrt{2}-2\left(\sqrt{2}+1\right)\right)\left(\sqrt{2}-2\right)}=y$$

Uża l-propjetà distributtiva biex timmultiplika -1+\sqrt{2} b'-2.

$$\frac{2-2\sqrt{2}}{\left(4+2\sqrt{2}-2\left(\sqrt{2}+1\right)\right)\left(\sqrt{2}-2\right)}=y$$

Uża l-propjetà distributtiva biex timmultiplika -2 b'\sqrt{2}+1.

$$\frac{2-2\sqrt{2}}{\left(4+2\sqrt{2}-2\sqrt{2}-2\right)\left(\sqrt{2}-2\right)}=y$$

Ikkombina 2\sqrt{2} u -2\sqrt{2} biex tikseb 0.

$$\frac{2-2\sqrt{2}}{\left(4-2\right)\left(\sqrt{2}-2\right)}=y$$

Naqqas 2 minn 4 biex tikseb 2.

$$\frac{2-2\sqrt{2}}{2\left(\sqrt{2}-2\right)}=y$$

Uża l-propjetà distributtiva biex timmultiplika 2 b'\sqrt{2}-2.

$$\frac{2-2\sqrt{2}}{2\sqrt{2}-4}=y$$

Irrazzjonalizza d-denominatur tal-\frac{2-2\sqrt{2}}{2\sqrt{2}-4} billi timmultiplika in-numeratur u d-denominatur mill-2\sqrt{2}+4.

$$\frac{\left(2-2\sqrt{2}\right)\left(2\sqrt{2}+4\right)}{\left(2\sqrt{2}-4\right)\left(2\sqrt{2}+4\right)}=y$$

Ikkunsidra li \left(2\sqrt{2}-4\right)\left(2\sqrt{2}+4\right). Il-multiplikazzjoni tista' tiġi ttrasformata fid-differenza tal-kwadrati li jużaw ir-regola: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(2-2\sqrt{2}\right)\left(2\sqrt{2}+4\right)}{\left(2\sqrt{2}\right)^{2}-4^{2}}=y$$

Espandi \left(2\sqrt{2}\right)^{2}.

$$\frac{\left(2-2\sqrt{2}\right)\left(2\sqrt{2}+4\right)}{2^{2}\left(\sqrt{2}\right)^{2}-4^{2}}=y$$

Ikkalkula 2 bil-power ta' 2 u tikseb 4.

$$\frac{\left(2-2\sqrt{2}\right)\left(2\sqrt{2}+4\right)}{4\left(\sqrt{2}\right)^{2}-4^{2}}=y$$

Il-kwadrat ta' \sqrt{2} huwa 2.

$$\frac{\left(2-2\sqrt{2}\right)\left(2\sqrt{2}+4\right)}{4\times 2-4^{2}}=y$$

Immultiplika 4 u 2 biex tikseb 8.

$$\frac{\left(2-2\sqrt{2}\right)\left(2\sqrt{2}+4\right)}{8-4^{2}}=y$$

Ikkalkula 4 bil-power ta' 2 u tikseb 16.

$$\frac{\left(2-2\sqrt{2}\right)\left(2\sqrt{2}+4\right)}{8-16}=y$$

Naqqas 16 minn 8 biex tikseb -8.

$$\frac{\left(2-2\sqrt{2}\right)\left(2\sqrt{2}+4\right)}{-8}=y$$

Ibdel in-naħat sabiex it-termini varjabbli kollha jkunu fuq in-naħa tax-xellug.

$$y=\frac{\left(2-2\sqrt{2}\right)\left(2\sqrt{2}+4\right)}{-8}$$

Uża l-propjetà distributtiva biex timmultiplika 2-2\sqrt{2} b'2\sqrt{2}+4 u kkombina termini simili.

$$y=\frac{-4\sqrt{2}+8-4\left(\sqrt{2}\right)^{2}}{-8}$$

Il-kwadrat ta' \sqrt{2} huwa 2.

$$y=\frac{-4\sqrt{2}+8-4\times 2}{-8}$$

Immultiplika -4 u 2 biex tikseb -8.

$$y=\frac{-4\sqrt{2}+8-8}{-8}$$

Naqqas 8 minn 8 biex tikseb 0.

$$y=\frac{-4\sqrt{2}}{-8}$$

Iddividi -4\sqrt{2} b'-8 biex tikseb\frac{1}{2}\sqrt{2}.

$$y=\frac{1}{2}\sqrt{2}$$

Is-sistema issa solvuta.

$$x=\sqrt{2}+1 y=\frac{1}{2}\sqrt{2}$$