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

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

Solvi għal x

$$x = \frac{43 \sqrt{2} - 7}{2} \approx 26.905591591$$

Passi biex Issolvi l-Ekwazzjoni Lineari

Ikkonverti 17 fi frazzjoni \frac{34}{2}.

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

Billi -\frac{95}{2} u \frac{34}{2} għandhom l-istess denominatur, żidhom billi żżid in-numeraturi tagħhom.

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

Żid -95 u 34 biex tikseb -61.

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

Żid 8 u 17 biex tikseb 25.

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

Iddividi x-\frac{61}{2} b'\frac{25-9\sqrt{2}}{2} billi timmultiplika x-\frac{61}{2} bir-reċiproku ta' \frac{25-9\sqrt{2}}{2}.

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

Irrazzjonalizza d-denominatur tal-\frac{\left(x-\frac{61}{2}\right)\times 2}{25-9\sqrt{2}} billi timmultiplika in-numeratur u d-denominatur mill-25+9\sqrt{2}.

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

Ikkunsidra li \left(25-9\sqrt{2}\right)\left(25+9\sqrt{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}.

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

Ikkalkula 25 bil-power ta' 2 u tikseb 625.

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

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

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

Ikkalkula -9 bil-power ta' 2 u tikseb 81.

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

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

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

Immultiplika 81 u 2 biex tikseb 162.

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

Naqqas 162 minn 625 biex tikseb 463.

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

Irrazzjonalizza d-denominatur tal-\frac{4-\sqrt{2}}{-3-\sqrt{2}} billi timmultiplika in-numeratur u d-denominatur mill--3+\sqrt{2}.

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

Ikkunsidra li \left(-3-\sqrt{2}\right)\left(-3+\sqrt{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}.

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

Ikkwadra -3. Ikkwadra \sqrt{2}.

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

Naqqas 2 minn 9 biex tikseb 7.

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

Uża l-propjetà distributtiva biex timmultiplika x-\frac{61}{2} b'2.

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

Annulla 2 u 2.

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

Applika l-propjetà distributtiva billi timmultiplika kull terminu ta' 2x-61 b'kull terminu ta' 25+9\sqrt{2}.

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

Applika l-propjetà distributtiva billi timmultiplika kull terminu ta' 4-\sqrt{2} b'kull terminu ta' -3+\sqrt{2}.

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

Ikkombina 4\sqrt{2} u 3\sqrt{2} biex tikseb 7\sqrt{2}.

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

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

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

Naqqas 2 minn -12 biex tikseb -14.

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

Iddividi kull terminu ta' -14+7\sqrt{2} b'7 biex tikseb-2+\sqrt{2}.

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

Immultiplika ż-żewġ naħat tal-ekwazzjoni b'463.

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

Żid 1525 maż-żewġ naħat.

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

Żid -926 u 1525 biex tikseb 599.

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

Żid 549\sqrt{2} maż-żewġ naħat.

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

Ikkombina 463\sqrt{2} u 549\sqrt{2} biex tikseb 1012\sqrt{2}.

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

Ikkombina t-termini kollha li fihom x.

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

L-ekwazzjoni hija f'forma standard.

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

Iddividi ż-żewġ naħat b'50+18\sqrt{2}.

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

Meta tiddividi b'50+18\sqrt{2} titneħħa l-multiplikazzjoni b'50+18\sqrt{2}.

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

Iddividi 599+1012\sqrt{2} b'50+18\sqrt{2}.

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