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

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

x, y खातीर सोडोवचें

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

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

सोडोवणी पांवडे

पयलें समिकरण विचारांत घेवचें. समिकरणात अचल संख्येची ज्ञात मानां रिगोवचीं.

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

2 मेळोवंक 1 आनी 1 ची बेरीज करची.

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

न्युमरेटर आनी डिनोमिनेटर \sqrt{2}-2 न गुणून \frac{2}{\sqrt{2}+2} चो डिनोमिनेटर रेशनलायझ तर्कसंगत करचो.

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

विचारांत घेयात \left(\sqrt{2}+2\right)\left(\sqrt{2}-2\right). नेम वापरून गुणाकार विभिन्न चवकोनांत रुपांतरण करूं येताः \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

\sqrt{2} वर्गमूळ. 2 वर्गमूळ.

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

-2 मेळोवंक 2 आनी 4 वजा करचे.

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

-2 आनी -2 रद्द करचें.

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

-\left(\sqrt{2}-2\right) च्या विरुध्दार्थी अंक \sqrt{2}-2 आसा.

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

-1 मेळोवंक 1 आनी 2 वजा करचे.

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

बायनोमियल प्रमेयाचो वापर करून \left(a+b\right)^{2}=a^{2}+2ab+b^{2} विस्तारावचें \left(\sqrt{2}+1\right)^{2}.

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

\sqrt{2} चो वर्ग 2 आसा.

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

3 मेळोवंक 2 आनी 1 ची बेरीज करची.

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

4 मेळोवंक 3 आनी 1 ची बेरीज करची.

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

2 मेळोवंक 1 आनी 1 ची बेरीज करची.

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

न्युमरेटर आनी डिनोमिनेटर \sqrt{2}-2 न गुणून \frac{4+2\sqrt{2}-2\left(\sqrt{2}+1\right)}{\sqrt{2}+2} चो डिनोमिनेटर रेशनलायझ तर्कसंगत करचो.

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

विचारांत घेयात \left(\sqrt{2}+2\right)\left(\sqrt{2}-2\right). नेम वापरून गुणाकार विभिन्न चवकोनांत रुपांतरण करूं येताः \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

\sqrt{2} वर्गमूळ. 2 वर्गमूळ.

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

-2 मेळोवंक 2 आनी 4 वजा करचे.

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

\frac{\left(4+2\sqrt{2}-2\left(\sqrt{2}+1\right)\right)\left(\sqrt{2}-2\right)}{-2} च्या पुरकाक -1+\sqrt{2} गुणून \frac{\left(4+2\sqrt{2}-2\left(\sqrt{2}+1\right)\right)\left(\sqrt{2}-2\right)}{-2} न -1+\sqrt{2} क भाग लावचो.

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

-2 न -1+\sqrt{2} गुणपाक विभाजक विशमाचो वापर करचो.

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

\sqrt{2}+1 न -2 गुणपाक विभाजक विशमाचो वापर करचो.

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

0 मेळोवंक 2\sqrt{2} आनी -2\sqrt{2} एकठांय करचें.

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

2 मेळोवंक 4 आनी 2 वजा करचे.

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

\sqrt{2}-2 न 2 गुणपाक विभाजक विशमाचो वापर करचो.

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

न्युमरेटर आनी डिनोमिनेटर 2\sqrt{2}+4 न गुणून \frac{2-2\sqrt{2}}{2\sqrt{2}-4} चो डिनोमिनेटर रेशनलायझ तर्कसंगत करचो.

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

विचारांत घेयात \left(2\sqrt{2}-4\right)\left(2\sqrt{2}+4\right). नेम वापरून गुणाकार विभिन्न चवकोनांत रुपांतरण करूं येताः \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

\left(2\sqrt{2}\right)^{2} विस्तारीत करचो.

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

4 मेळोवंक 2 चो 2 पॉवर मेजचो.

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

\sqrt{2} चो वर्ग 2 आसा.

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

8 मेळोवंक 4 आनी 2 गुणचें.

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

16 मेळोवंक 2 चो 4 पॉवर मेजचो.

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

-8 मेळोवंक 8 आनी 16 वजा करचे.

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

कुशी हाणच्यो ताका लागून बरोबर चिन्नाच्या दाव्यान सगळी विशम संज्ञा येतली.

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

वितरक गूणधर्माचो वापर करून 2-2\sqrt{2} क 2\sqrt{2}+4 न गुणचें आनी संज्ञां भशेन एकठावणी करची.

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

\sqrt{2} चो वर्ग 2 आसा.

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

-8 मेळोवंक -4 आनी 2 गुणचें.

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

0 मेळोवंक 8 आनी 8 वजा करचे.

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

\frac{1}{2}\sqrt{2} मेळोवंक -4\sqrt{2} क -8 न भाग लावचो.

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

प्रणाली आतां सुटावी जाली.

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