Nafaqada saqda dhexe waxay magaceeda ka helaysaa shuruuda dugsiga: waa inaad akhrin kartaa qaacidada xalka ee isla'egyada quadratic xiitaa haddii lagu toosiyo saqda dhexe habeenkii. Darajooyinka saddexaad iyo afraad, ciyaarta fikirkani waxa ay noqotaa mid aad u baahi badan. Waxa kale oo jira habab xidhan oo tan loo isticmaalayo xisaabinta aasaasiga ah iyo xididdada, laakiin aad bay u dheer yihiin. Halkan waxay si buuxda ugu jiraan isku-beegyada asalka ah \(a,b,c,d,e\) , iyada oo aan lahayn doorsoomayaal caawiye ah oo leh shuruudaha lagama maarmaanka ah.
Iskuxidhka hogaamineed wuxuu had iyo jeer qanciyaa \(a\ne0\) ; haddii kale shahaadadu hoos u dhacdo. Waxaan raadineynaa dhammaan eber adag oo ay ku jiraan isku dhufashada. Kiis toosan \(n=1\) iskuxirayaasha \(a\) iyo \(b\) ayaa ku filan: isla'egta waxaa si toos ah loogu xalin karaa \(x\) oo waxay leedahay hal eber oo fudud. Tani waxay sidoo kale khuseysaa iskuxirayaasha isku dhafan. Wixii \(a=0\) si kastaba ha ahaatee, ma jiro wax xal ah \(b\ne0\) ; waa \(a=b=0\) , kakan kasta \(x\) waa xal. Isla'egta toosan ee dhabta ah waa la helayaa:
$$ax+b=0,\qquad a\ne0.$$
$$\boxed{x_k=-\frac{b}{a},\qquad k\in\{1\}.}$$
Waayo \(n=2\) isla'egta quadratic waxay ka kooban tahay iskuxirayaasha \(a,b,c\) . Iyadoo \(k=1,2\) \((-1)^k\) beddesho calaamada oo soo celisa labada eber oo ay ku jiraan dhufashada. Isku-dhafka dhabta ah waxaa jira laba xal oo kala duwan oo dhab ah oo loogu talagalay \(b^2-4ac>0\) , loogu talagalay \(b^2-4ac=0\) laba-laaban iyo qiimo taban laba isku-dhafan: Kadib waxaad dooran kartaa xididdada si \(\sqrt{b^2-4ac}=i\sqrt{4ac-b^2}\) \(i\) \(i^2=-1\) waa. Ma jiro wax kala soocida kiis isugeynta kakan; qaacidada waxay quseysaa labada xidid ee labajibbaaran:
$$ax^2+bx+c=0,\qquad a\ne0.$$
$$\boxed{x_k=\frac{-b+(-1)^k\sqrt{b^2-4ac}}{2a},\qquad k\in\{1,2\}.}$$
Waayo, \(n=3\) Qaaciddada Cardano waxay si toos ah ugu qoran tahay \(a,b,c,d\) . Xarafka \(k\) waa kaliya tusaha xalka: qiyamka \(1,2,3\) waxay doortaan saddexda eber, ma jiraan tiro kale oo caawiye ah. Awooddu waxay ka kooban tahay asal saddexaad oo asal ah oo midnimada . Astaan kasta oo xidid ah waxay u taagan tahay xidid kasta oo adag; tibaaxaha soo noqnoqda waa in ay tilmaamayaan isla qiimaha meel kasta. Kaliya oo leh \(b^2-3ac=0\) waa in xididka labajibbaaran ee gudaha uu leeyahay qiimaha \(2b^3-9abc+27a^2d\) , haddii kale radicand xididka cube wuxuu noqonayaa eber.:
$$ax^3+bx^2+cx+d=0,\qquad a\ne0.$$
$$\left(b^2-3ac,\;2b^3-9abc+27a^2d\right)\ne(0,0).$$
$$\boxed{\begin{gathered}x_k=-\frac{b}{3a}-\frac{\left(\frac{-1+i\sqrt{3}}{2}\right)^{k-1}}{3a}\sqrt[3]{\frac{1}{2}\left(\begin{aligned}&2b^3-9abc+27a^2d\\&+\sqrt{\left(2b^3-9abc+27a^2d\right)^2-4\left(b^2-3ac\right)^3}\end{aligned}\right)}-\frac{b^2-3ac}{3a\left(\frac{-1+i\sqrt{3}}{2}\right)^{k-1}\sqrt[3]{\frac{1}{2}\left(\begin{aligned}&2b^3-9abc+27a^2d\\&+\sqrt{\left(2b^3-9abc+27a^2d\right)^2-4\left(b^2-3ac\right)^3}\end{aligned}\right)}}\\[5pt]k\in\{1,2,3\}.\end{gathered}}$$
\(x^3-1=0\) waxay tusinaysaa sida go'aaminta xididka labajibbaaran ee gudaha u shaqeeyo: Halkan \(b^2-3ac=0\) iyo \(2b^3-9abc+27a^2d=-27\) . Marka la eego qiimaha \(27\) halkii \(-27\) xididku wuxuu noqon lahaa \(\frac{-27+27}{2}=0\) oo qaacidadu ma noqon doonto mid la qiimeeyo. Xaaladda ku jirta isla'egta hore waxay meesha ka saaraysaa sida saxda ah ee ay labada erayba eber yihiin iyo xididka cube sidaas darteed mar walba wuu baaba'aa. Isla'egta ayaa markaa leh eber saddex-laab ah oo aan lagu daboolin qaacidada halkan lagu sheegay.
Isku-dhafka cubic-ka dhabta ah, calaamadda \(b^2c^2-4ac^3-4b^3d-27a^2d^2+18abcd\) ayaa go'aamisa eber: Qiimaha togan macnaheedu waa saddex xalal dhab ah oo kala duwan, qiimaha taban macnaheedu waa mid dhab ah iyo laba aan dhab ahayn, eber dhammaan waa dhab oo ku celcelin ah. Xaaladda kowaad, qaaciddada Cardano waxay ku qasbeysaa qiyamka dhexdhexaadka ah ee adag. Qaababka xagjirka ah aljabra ahaan waa sax, laakiin had iyo jeer ma aha tiro ahaan xasilloon: ku dhawaad qiyamyada dhexdhexaadka ah ee isku midka ah waxay ku kici karaan tirooyin xisaabinta dhibcaha sabbaynaya; haddii loo baahdo, saxsanaan sare ayaa la isticmaalaa.
Waayo, \(n=4\) Habka Ferrari waxa kale oo si buuxda loogu qoray \(a,b,c,d,e\) . Tusaha \(k\) wuxuu ka bilaabmaa \(1\) ilaa \(4\) ; Awoodaha \(-1\) waxay qabtaan dhammaan afarta calaamadood ee isku-dhafka ah waxayna si habboon u lamaanaan xididdada. Tibaaxaha asalka ah ee isku midka ah ayaa tilmaamaya isla qiimaha, xitaa noocyada kala duwan \(k\) . Waayo \(c^2-3bd+12ae=0\) \(2c^3-9bcd+27b^2e+27ad^2-72ace\) waa qiimaha xididka labajibbaaran ee gudaha. Xaaladaha hoos yimaada isla'egta hore waxay ka hortagayaan u qaybinta eber; eber kasta waxaa lagu duubayaa badnaantiisa:
$$ax^4+bx^3+cx^2+dx+e=0,\qquad a\ne0.$$
$$\begin{gathered}b^3-4abc+8a^2d\ne0,\\\left(c^2-3bd+12ae,\;2c^3-9bcd+27b^2e+27ad^2-72ace\right)\ne(0,0).\end{gathered}$$
$$\boxed{\begin{gathered}x_k=-\frac{b}{4a}\\[6pt]+\frac{(-1)^{\frac{k(k-1)}{2}}\sqrt{3}}{12a}\sqrt{\begin{aligned}&3b^2-8ac+4a\sqrt[3]{\frac{1}{2}\left(2c^3-9bcd+27b^2e+27ad^2-72ace+\sqrt{\left(2c^3-9bcd+27b^2e+27ad^2-72ace\right)^2-4\left(c^2-3bd+12ae\right)^3}\right)}\\&+\frac{4a\left(c^2-3bd+12ae\right)}{\sqrt[3]{\frac{1}{2}\left(2c^3-9bcd+27b^2e+27ad^2-72ace+\sqrt{\left(2c^3-9bcd+27b^2e+27ad^2-72ace\right)^2-4\left(c^2-3bd+12ae\right)^3}\right)}}\end{aligned}}\\[6pt]+\frac{(-1)^k\sqrt{3}}{12a}\sqrt{\begin{aligned}&6b^2-16ac-4a\sqrt[3]{\frac{1}{2}\left(2c^3-9bcd+27b^2e+27ad^2-72ace+\sqrt{\left(2c^3-9bcd+27b^2e+27ad^2-72ace\right)^2-4\left(c^2-3bd+12ae\right)^3}\right)}\\&-\frac{4a\left(c^2-3bd+12ae\right)}{\sqrt[3]{\frac{1}{2}\left(2c^3-9bcd+27b^2e+27ad^2-72ace+\sqrt{\left(2c^3-9bcd+27b^2e+27ad^2-72ace\right)^2-4\left(c^2-3bd+12ae\right)^3}\right)}}\\&-\frac{6(-1)^{\frac{k(k-1)}{2}}\sqrt{3}\left(b^3-4abc+8a^2d\right)}{\sqrt{\begin{aligned}&3b^2-8ac+4a\sqrt[3]{\frac{1}{2}\left(2c^3-9bcd+27b^2e+27ad^2-72ace+\sqrt{\left(2c^3-9bcd+27b^2e+27ad^2-72ace\right)^2-4\left(c^2-3bd+12ae\right)^3}\right)}\\&+\frac{4a\left(c^2-3bd+12ae\right)}{\sqrt[3]{\frac{1}{2}\left(2c^3-9bcd+27b^2e+27ad^2-72ace+\sqrt{\left(2c^3-9bcd+27b^2e+27ad^2-72ace\right)^2-4\left(c^2-3bd+12ae\right)^3}\right)}}\end{aligned}}}\end{aligned}}\\[6pt]k\in\{1,2,3,4\}.\end{gathered}}$$
Xaaladda kowaad waxa ka saaraysa halbeegyo isku mid ah oo ku saabsan booska \(x=-\frac{b}{4a}\) , oo ay ku jiraan isla'egyada labajibbaaran ee \(b=d=0\) sida \(x^4-5x^2+4=0\) . Halkaa xididka weyni wuxuu noqon karaa eber ka dibna wuxuu ku jiraa hooseeyaha. Xaaladda labaad waxa ay meesha ka saaraysaa ugu yaraan eber saddex jibbaaran, tusaale ahaan \(x^4-6x^2+8x-3=(x-1)^3(x+3)=0\) : Halkan xididka cube-ga ayaa ku dhex baaba'aya tirada. Kiisaska gaarka ah ee laga saaray kuma jiraan caanaha sanduuqa; Hababka xalinta dheeraadka ah ayaa halkan laga saaray.
Laga soo bilaabo \(n=5\) xitaa xirfadaha xifdiga ugu fiican waxba ma caawiyaan: ma jiro qaacido guud oo ka kooban tiro xaddidan oo ah hawlgallada xisaabinta aasaasiga ah iyo xididada. Tani waa waxa aragtida Haabiil-Ruffini yidhi; caddaynta waxaa laga heli karaa Milne's Fields iyo Aragtida Galois . Isla'egyada shaqsiga ah ayaa wali lagu xalin karaa iyadoo la adeegsanayo xagjir. Iyo qiyamka kale ee la qiyaasi karo iyo hawlaha gaarka ah ayaa weli ah, sida Keenka xagjirka ah ee darajada shanaad. Markaa eberku ma baaba'aan, kaliya rajada xidid caalami ah.