Riješi x-95/2+17/8-9sqrt{2+17/2}=frac{4-sqrt{2}}{-3-sqrt{2}}

$$\frac { x - \frac { 95 } { 2 } + 17 } { \frac { 8 - 9 \sqrt { 2 } + 17 } { 2 } } = \frac { 4 - \sqrt { 2 } } { - 3 - \sqrt { 2 } }$$

Riješite za x

$$x = \frac{43 \sqrt{2} - 7}{2} \approx 26,905591591$$

Koraci za rješavanje linearne jednačine

Konvertirajte 17 u razlomak \frac{34}{2}.

$$\frac{x-\frac{95}{2}+\frac{34}{2}}{\frac{8-9\sqrt{2}+17}{2}}=\frac{4-\sqrt{2}}{-3-\sqrt{2}}$$

Pošto -\frac{95}{2} i \frac{34}{2} imaju isti imenilac, saberite ih tako što ćete sabrati njihove brojioce.

$$\frac{x+\frac{-95+34}{2}}{\frac{8-9\sqrt{2}+17}{2}}=\frac{4-\sqrt{2}}{-3-\sqrt{2}}$$

Saberite -95 i 34 da biste dobili -61.

$$\frac{x-\frac{61}{2}}{\frac{8-9\sqrt{2}+17}{2}}=\frac{4-\sqrt{2}}{-3-\sqrt{2}}$$

Saberite 8 i 17 da biste dobili 25.

$$\frac{x-\frac{61}{2}}{\frac{25-9\sqrt{2}}{2}}=\frac{4-\sqrt{2}}{-3-\sqrt{2}}$$

Podijelite x-\frac{61}{2} sa \frac{25-9\sqrt{2}}{2} tako što ćete pomnožiti x-\frac{61}{2} recipročnom vrijednošću od \frac{25-9\sqrt{2}}{2}.

$$\frac{\left(x-\frac{61}{2}\right)\times 2}{25-9\sqrt{2}}=\frac{4-\sqrt{2}}{-3-\sqrt{2}}$$

Racionalizirajte imenilac broja \frac{\left(x-\frac{61}{2}\right)\times 2}{25-9\sqrt{2}} tako što ćete pomnožiti brojilac i imenilac sa 25+9\sqrt{2}.

$$\frac{\left(x-\frac{61}{2}\right)\times 2\left(25+9\sqrt{2}\right)}{\left(25-9\sqrt{2}\right)\left(25+9\sqrt{2}\right)}=\frac{4-\sqrt{2}}{-3-\sqrt{2}}$$

Razmotrite \left(25-9\sqrt{2}\right)\left(25+9\sqrt{2}\right). Množenje se može transformirati u razliku kvadrata pomoću pravila: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(x-\frac{61}{2}\right)\times 2\left(25+9\sqrt{2}\right)}{25^{2}-\left(-9\sqrt{2}\right)^{2}}=\frac{4-\sqrt{2}}{-3-\sqrt{2}}$$

Izračunajte 25 stepen od 2 i dobijte 625.

$$\frac{\left(x-\frac{61}{2}\right)\times 2\left(25+9\sqrt{2}\right)}{625-\left(-9\sqrt{2}\right)^{2}}=\frac{4-\sqrt{2}}{-3-\sqrt{2}}$$

Proširite \left(-9\sqrt{2}\right)^{2}.

$$\frac{\left(x-\frac{61}{2}\right)\times 2\left(25+9\sqrt{2}\right)}{625-\left(-9\right)^{2}\left(\sqrt{2}\right)^{2}}=\frac{4-\sqrt{2}}{-3-\sqrt{2}}$$

Izračunajte -9 stepen od 2 i dobijte 81.

$$\frac{\left(x-\frac{61}{2}\right)\times 2\left(25+9\sqrt{2}\right)}{625-81\left(\sqrt{2}\right)^{2}}=\frac{4-\sqrt{2}}{-3-\sqrt{2}}$$

Kvadrat broja \sqrt{2} je 2.

$$\frac{\left(x-\frac{61}{2}\right)\times 2\left(25+9\sqrt{2}\right)}{625-81\times 2}=\frac{4-\sqrt{2}}{-3-\sqrt{2}}$$

Pomnožite 81 i 2 da biste dobili 162.

$$\frac{\left(x-\frac{61}{2}\right)\times 2\left(25+9\sqrt{2}\right)}{625-162}=\frac{4-\sqrt{2}}{-3-\sqrt{2}}$$

Oduzmite 162 od 625 da biste dobili 463.

$$\frac{\left(x-\frac{61}{2}\right)\times 2\left(25+9\sqrt{2}\right)}{463}=\frac{4-\sqrt{2}}{-3-\sqrt{2}}$$

Racionalizirajte imenilac broja \frac{4-\sqrt{2}}{-3-\sqrt{2}} tako što ćete pomnožiti brojilac i imenilac sa -3+\sqrt{2}.

$$\frac{\left(x-\frac{61}{2}\right)\times 2\left(25+9\sqrt{2}\right)}{463}=\frac{\left(4-\sqrt{2}\right)\left(-3+\sqrt{2}\right)}{\left(-3-\sqrt{2}\right)\left(-3+\sqrt{2}\right)}$$

Razmotrite \left(-3-\sqrt{2}\right)\left(-3+\sqrt{2}\right). Množenje se može transformirati u razliku kvadrata pomoću pravila: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(x-\frac{61}{2}\right)\times 2\left(25+9\sqrt{2}\right)}{463}=\frac{\left(4-\sqrt{2}\right)\left(-3+\sqrt{2}\right)}{\left(-3\right)^{2}-\left(\sqrt{2}\right)^{2}}$$

Izračunajte kvadrat od -3. Izračunajte kvadrat od \sqrt{2}.

$$\frac{\left(x-\frac{61}{2}\right)\times 2\left(25+9\sqrt{2}\right)}{463}=\frac{\left(4-\sqrt{2}\right)\left(-3+\sqrt{2}\right)}{9-2}$$

Oduzmite 2 od 9 da biste dobili 7.

$$\frac{\left(x-\frac{61}{2}\right)\times 2\left(25+9\sqrt{2}\right)}{463}=\frac{\left(4-\sqrt{2}\right)\left(-3+\sqrt{2}\right)}{7}$$

Koristite distributivno svojstvo da biste pomnožili x-\frac{61}{2} sa 2.

$$\frac{\left(2x-\frac{61}{2}\times 2\right)\left(25+9\sqrt{2}\right)}{463}=\frac{\left(4-\sqrt{2}\right)\left(-3+\sqrt{2}\right)}{7}$$

Otkaži 2 i 2.

$$\frac{\left(2x-61\right)\left(25+9\sqrt{2}\right)}{463}=\frac{\left(4-\sqrt{2}\right)\left(-3+\sqrt{2}\right)}{7}$$

Primijenite distributivno svojstvo tako što ćete pomnožiti svaki izraz od 2x-61 svakim izrazom od 25+9\sqrt{2}.

$$\frac{50x+18\sqrt{2}x-1525-549\sqrt{2}}{463}=\frac{\left(4-\sqrt{2}\right)\left(-3+\sqrt{2}\right)}{7}$$

Primijenite distributivno svojstvo tako što ćete pomnožiti svaki izraz od 4-\sqrt{2} svakim izrazom od -3+\sqrt{2}.

$$\frac{50x+18\sqrt{2}x-1525-549\sqrt{2}}{463}=\frac{-12+4\sqrt{2}+3\sqrt{2}-\left(\sqrt{2}\right)^{2}}{7}$$

Kombinirajte 4\sqrt{2} i 3\sqrt{2} da biste dobili 7\sqrt{2}.

$$\frac{50x+18\sqrt{2}x-1525-549\sqrt{2}}{463}=\frac{-12+7\sqrt{2}-\left(\sqrt{2}\right)^{2}}{7}$$

Kvadrat broja \sqrt{2} je 2.

$$\frac{50x+18\sqrt{2}x-1525-549\sqrt{2}}{463}=\frac{-12+7\sqrt{2}-2}{7}$$

Oduzmite 2 od -12 da biste dobili -14.

$$\frac{50x+18\sqrt{2}x-1525-549\sqrt{2}}{463}=\frac{-14+7\sqrt{2}}{7}$$

Podijelite svaki element izraza -14+7\sqrt{2} s 7 da biste dobili -2+\sqrt{2}.

$$\frac{50x+18\sqrt{2}x-1525-549\sqrt{2}}{463}=-2+\sqrt{2}$$

Pomnožite obje strane jednačine sa 463.

$$50x+18\sqrt{2}x-1525-549\sqrt{2}=-926+463\sqrt{2}$$

Dodajte 1525 na obje strane.

$$50x+18\sqrt{2}x-549\sqrt{2}=-926+463\sqrt{2}+1525$$

Saberite -926 i 1525 da biste dobili 599.

$$50x+18\sqrt{2}x-549\sqrt{2}=599+463\sqrt{2}$$

Dodajte 549\sqrt{2} na obje strane.

$$50x+18\sqrt{2}x=599+463\sqrt{2}+549\sqrt{2}$$

Kombinirajte 463\sqrt{2} i 549\sqrt{2} da biste dobili 1012\sqrt{2}.

$$50x+18\sqrt{2}x=599+1012\sqrt{2}$$

Kombinirajte sve termine koji sadrže x.

$$\left(50+18\sqrt{2}\right)x=599+1012\sqrt{2}$$

Jednačina je u standardnom obliku.

$$\left(18\sqrt{2}+50\right)x=1012\sqrt{2}+599$$

Podijelite obje strane s 50+18\sqrt{2}.

$$\frac{\left(18\sqrt{2}+50\right)x}{18\sqrt{2}+50}=\frac{1012\sqrt{2}+599}{18\sqrt{2}+50}$$

Dijelјenje sa 50+18\sqrt{2} poništava množenje sa 50+18\sqrt{2}.

$$x=\frac{1012\sqrt{2}+599}{18\sqrt{2}+50}$$

Podijelite 599+1012\sqrt{2} sa 50+18\sqrt{2}.

$$x=\frac{43\sqrt{2}-7}{2}$$