Issolvi 8y-5/2y+5=53y+7/-3y+2

$$\frac{ 8y-5 }{ 2y+5 } = 5 \frac{ 3y+7 }{ -3y+2 }$$

Solvi għal y

$$y=\frac{-\sqrt{749}i-19}{18}\approx -1.055555556-1.520436909i$$

$$y=\frac{-19+\sqrt{749}i}{18}\approx -1.055555556+1.520436909i$$

Passi għall-Użu tal-Formula Kwadratika

Il-varjabbli y ma jistax ikun ugwali għal kwalunkwe mill-valuri -\frac{5}{2},\frac{2}{3} billi d-diviżjoni b'żero mhix definita. Immultiplika ż-żewġ naħat tal-ekwazzjoni b'\left(3y-2\right)\left(2y+5\right), l-inqas denominatur komuni ta' 2y+5,-3y+2.

$$\left(3y-2\right)\left(8y-5\right)=5\left(-5-2y\right)\left(3y+7\right)$$

Uża l-propjetà distributtiva biex timmultiplika 3y-2 b'8y-5 u kkombina termini simili.

$$24y^{2}-31y+10=5\left(-5-2y\right)\left(3y+7\right)$$

Uża l-propjetà distributtiva biex timmultiplika 5 b'-5-2y.

$$24y^{2}-31y+10=\left(-25-10y\right)\left(3y+7\right)$$

Uża l-propjetà distributtiva biex timmultiplika -25-10y b'3y+7 u kkombina termini simili.

$$24y^{2}-31y+10=-145y-175-30y^{2}$$

Żid 145y maż-żewġ naħat.

$$24y^{2}-31y+10+145y=-175-30y^{2}$$

Ikkombina -31y u 145y biex tikseb 114y.

$$24y^{2}+114y+10=-175-30y^{2}$$

Naqqas -175 miż-żewġ naħat.

$$24y^{2}+114y+10-\left(-175\right)=-30y^{2}$$

L-oppost ta' -175 huwa 175.

$$24y^{2}+114y+10+175=-30y^{2}$$

Żid 30y^{2} maż-żewġ naħat.

$$24y^{2}+114y+10+175+30y^{2}=0$$

Żid 10 u 175 biex tikseb 185.

$$24y^{2}+114y+185+30y^{2}=0$$

Ikkombina 24y^{2} u 30y^{2} biex tikseb 54y^{2}.

$$54y^{2}+114y+185=0$$

Din l-ekwazzjoni hija fil-forma standard: ax^{2}+bx+c=0. Issostitwixxi 54 għal a, 114 għal b, u 185 għal c fil-formula kwadratika, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-114±\sqrt{114^{2}-4\times 54\times 185}}{2\times 54}$$

Ikkwadra 114.

$$y=\frac{-114±\sqrt{12996-4\times 54\times 185}}{2\times 54}$$

Immultiplika -4 b'54.

$$y=\frac{-114±\sqrt{12996-216\times 185}}{2\times 54}$$

Immultiplika -216 b'185.

$$y=\frac{-114±\sqrt{12996-39960}}{2\times 54}$$

Żid 12996 ma' -39960.

$$y=\frac{-114±\sqrt{-26964}}{2\times 54}$$

Ħu l-għerq kwadrat ta' -26964.

$$y=\frac{-114±6\sqrt{749}i}{2\times 54}$$

Immultiplika 2 b'54.

$$y=\frac{-114±6\sqrt{749}i}{108}$$

Issa solvi l-ekwazzjoni y=\frac{-114±6\sqrt{749}i}{108} fejn ± hija plus. Żid -114 ma' 6i\sqrt{749}.

$$y=\frac{-114+6\sqrt{749}i}{108}$$

Iddividi -114+6i\sqrt{749} b'108.

$$y=\frac{-19+\sqrt{749}i}{18}$$

Issa solvi l-ekwazzjoni y=\frac{-114±6\sqrt{749}i}{108} fejn ± hija minus. Naqqas 6i\sqrt{749} minn -114.

$$y=\frac{-6\sqrt{749}i-114}{108}$$

Iddividi -114-6i\sqrt{749} b'108.

$$y=\frac{-\sqrt{749}i-19}{18}$$

L-ekwazzjoni issa solvuta.

$$y=\frac{-19+\sqrt{749}i}{18} y=\frac{-\sqrt{749}i-19}{18}$$

Passi biex Tikkompleta l-Kwadrat

Il-varjabbli y ma jistax ikun ugwali għal kwalunkwe mill-valuri -\frac{5}{2},\frac{2}{3} billi d-diviżjoni b'żero mhix definita. Immultiplika ż-żewġ naħat tal-ekwazzjoni b'\left(3y-2\right)\left(2y+5\right), l-inqas denominatur komuni ta' 2y+5,-3y+2.

$$\left(3y-2\right)\left(8y-5\right)=5\left(-5-2y\right)\left(3y+7\right)$$

Uża l-propjetà distributtiva biex timmultiplika 3y-2 b'8y-5 u kkombina termini simili.

$$24y^{2}-31y+10=5\left(-5-2y\right)\left(3y+7\right)$$

Uża l-propjetà distributtiva biex timmultiplika 5 b'-5-2y.

$$24y^{2}-31y+10=\left(-25-10y\right)\left(3y+7\right)$$

Uża l-propjetà distributtiva biex timmultiplika -25-10y b'3y+7 u kkombina termini simili.

$$24y^{2}-31y+10=-145y-175-30y^{2}$$

Żid 145y maż-żewġ naħat.

$$24y^{2}-31y+10+145y=-175-30y^{2}$$

Ikkombina -31y u 145y biex tikseb 114y.

$$24y^{2}+114y+10=-175-30y^{2}$$

Żid 30y^{2} maż-żewġ naħat.

$$24y^{2}+114y+10+30y^{2}=-175$$

Ikkombina 24y^{2} u 30y^{2} biex tikseb 54y^{2}.

$$54y^{2}+114y+10=-175$$

Naqqas 10 miż-żewġ naħat.

$$54y^{2}+114y=-175-10$$

Naqqas 10 minn -175 biex tikseb -185.

$$54y^{2}+114y=-185$$

Iddividi ż-żewġ naħat b'54.

$$\frac{54y^{2}+114y}{54}=-\frac{185}{54}$$

Meta tiddividi b'54 titneħħa l-multiplikazzjoni b'54.

$$y^{2}+\frac{114}{54}y=-\frac{185}{54}$$

Naqqas il-frazzjoni \frac{114}{54} għat-termini l-aktar baxxi billi testratta u tikkanċella barra 6.

$$y^{2}+\frac{19}{9}y=-\frac{185}{54}$$

Iddividi \frac{19}{9}, il-koeffiċjent tat-terminu x, b'2 biex tikseb \frac{19}{18}. Imbagħad żid il-kwadru ta' \frac{19}{18} maż-żewġ naħat tal-ekwazzjoni. Dan il-pass jagħmel in-naħa tax-xellug tal-ekwazzjoni kwadru perfett.

$$y^{2}+\frac{19}{9}y+\left(\frac{19}{18}\right)^{2}=-\frac{185}{54}+\left(\frac{19}{18}\right)^{2}$$

Ikkwadra \frac{19}{18} billi tikkwadra kemm in-numeratur u d-denominatur tal-frazzjoni.

$$y^{2}+\frac{19}{9}y+\frac{361}{324}=-\frac{185}{54}+\frac{361}{324}$$

Żid -\frac{185}{54} ma' \frac{361}{324} biex issib id-denominatur komuni u żżid in-numeraturi. Imbagħad naqqas il-frazzjoni għat-termini l-aktar baxxi jekk possibbli.

$$y^{2}+\frac{19}{9}y+\frac{361}{324}=-\frac{749}{324}$$

Fattur y^{2}+\frac{19}{9}y+\frac{361}{324}. B'mod ġenerali, meta x^{2}+bx+c huwa kwadru perfett, dejjem jista' jiġu fatturati bħala \left(x+\frac{b}{2}\right)^{2}.

$$\left(y+\frac{19}{18}\right)^{2}=-\frac{749}{324}$$

Ħu l-għerq kwadrat taż-żewġ naħat tal-ekwazzjoni.

$$\sqrt{\left(y+\frac{19}{18}\right)^{2}}=\sqrt{-\frac{749}{324}}$$

Issimplifika.

$$y+\frac{19}{18}=\frac{\sqrt{749}i}{18} y+\frac{19}{18}=-\frac{\sqrt{749}i}{18}$$

Naqqas \frac{19}{18} miż-żewġ naħat tal-ekwazzjoni.

$$y=\frac{-19+\sqrt{749}i}{18} y=\frac{-\sqrt{749}i-19}{18}$$