Réitigh sqrt{4578}x^2-sqrt{4677521}x+31478=10523

$$\sqrt{ 4578 } { x }^{ 2 } - \sqrt{ 4677521 } x+31478=10523$$

Réitigh do x. (complex solution)

$$x=\frac{\sqrt{21413691138}+i\sqrt{383727960\sqrt{4578}-21413691138}}{9156}\approx 15.982325934+7.366910212i$$

$$x=\frac{-i\sqrt{383727960\sqrt{4578}-21413691138}+\sqrt{21413691138}}{9156}\approx 15.982325934-7.366910212i$$

Céimeanna ag Baint Úsáid as an bhFoirmle Chearnach

Bain 10523 ón dá thaobh.

$$\sqrt{4578}x^{2}-\sqrt{4677521}x+31478-10523=0$$

Dealaigh 10523 ó 31478 chun 20955 a fháil.

$$\sqrt{4578}x^{2}-\sqrt{4677521}x+20955=0$$

Is féidir gach cothromóid san fhoirm ax^{2}+bx+c=0 a réiteach ag baint úsáid as an bhfoirmle chearnach : \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Tugann an fhoirmle chearnach dhá réiteach, ceann amháin nuair is suimiú é ± agus ceann eile nuair is dealú é.

$$\sqrt{4578}x^{2}+\left(-\sqrt{4677521}\right)x+20955=0$$

Tá an chothromóid seo i bhfoirm chaighdeánach: ax^{2}+bx+c=0. Cuir \sqrt{4578} in ionad a, -\sqrt{4677521} in ionad b, agus 20955 in ionad c san fhoirmle chearnach, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\sqrt{4677521}\right)±\sqrt{\left(-\sqrt{4677521}\right)^{2}-4\sqrt{4578}\times 20955}}{2\sqrt{4578}}$$

Cearnóg -\sqrt{4677521}.

$$x=\frac{-\left(-\sqrt{4677521}\right)±\sqrt{4677521-4\sqrt{4578}\times 20955}}{2\sqrt{4578}}$$

Méadaigh -4 faoi \sqrt{4578}.

$$x=\frac{-\left(-\sqrt{4677521}\right)±\sqrt{4677521+\left(-4\sqrt{4578}\right)\times 20955}}{2\sqrt{4578}}$$

Méadaigh -4\sqrt{4578} faoi 20955.

$$x=\frac{-\left(-\sqrt{4677521}\right)±\sqrt{4677521-83820\sqrt{4578}}}{2\sqrt{4578}}$$

Tóg fréamh chearnach 4677521-83820\sqrt{4578}.

$$x=\frac{-\left(-\sqrt{4677521}\right)±i\sqrt{-\left(4677521-83820\sqrt{4578}\right)}}{2\sqrt{4578}}$$

Tá \sqrt{4677521} urchomhairleach le -\sqrt{4677521}.

$$x=\frac{\sqrt{4677521}±i\sqrt{-\left(4677521-83820\sqrt{4578}\right)}}{2\sqrt{4578}}$$

Réitigh an chothromóid x=\frac{\sqrt{4677521}±i\sqrt{-\left(4677521-83820\sqrt{4578}\right)}}{2\sqrt{4578}} nuair is ionann ± agus plus. Suimigh \sqrt{4677521} le i\sqrt{-\left(4677521-83820\sqrt{4578}\right)}?

$$x=\frac{\sqrt{4677521}+i\sqrt{83820\sqrt{4578}-4677521}}{2\sqrt{4578}}$$

Roinn \sqrt{4677521}+i\sqrt{-4677521+83820\sqrt{4578}} faoi 2\sqrt{4578}.

$$x=\frac{\sqrt{4578}\left(\sqrt{4677521}+i\sqrt{83820\sqrt{4578}-4677521}\right)}{9156}$$

Réitigh an chothromóid x=\frac{\sqrt{4677521}±i\sqrt{-\left(4677521-83820\sqrt{4578}\right)}}{2\sqrt{4578}} nuair is ionann ± agus míneas. Dealaigh i\sqrt{-\left(4677521-83820\sqrt{4578}\right)} ó \sqrt{4677521}.

$$x=\frac{-i\sqrt{83820\sqrt{4578}-4677521}+\sqrt{4677521}}{2\sqrt{4578}}$$

Roinn \sqrt{4677521}-i\sqrt{-4677521+83820\sqrt{4578}} faoi 2\sqrt{4578}.

$$x=\frac{\sqrt{4578}\left(-i\sqrt{83820\sqrt{4578}-4677521}+\sqrt{4677521}\right)}{9156}$$

Tá an chothromóid réitithe anois.

$$x=\frac{\sqrt{4578}\left(\sqrt{4677521}+i\sqrt{83820\sqrt{4578}-4677521}\right)}{9156} x=\frac{\sqrt{4578}\left(-i\sqrt{83820\sqrt{4578}-4677521}+\sqrt{4677521}\right)}{9156}$$

Céimeanna chun an Chearnóg a Chomhlánú

Bain 31478 ón dá thaobh.

$$\sqrt{4578}x^{2}-\sqrt{4677521}x=10523-31478$$

Dealaigh 31478 ó 10523 chun -20955 a fháil.

$$\sqrt{4578}x^{2}-\sqrt{4677521}x=-20955$$

Is féidir cothromóidí cearnach cosúil leis an gceann seo a réitigh tríd an gcearnóg a chomhlánú. Chun an chearnóg a chomhlánú, ní mór don chothromóid a bheith san fhoirm x^{2}+bx=c ar dtús.

$$\sqrt{4578}x^{2}+\left(-\sqrt{4677521}\right)x=-20955$$

Roinn an dá thaobh faoi \sqrt{4578}.

$$\frac{\sqrt{4578}x^{2}+\left(-\sqrt{4677521}\right)x}{\sqrt{4578}}=-\frac{20955}{\sqrt{4578}}$$

Má roinntear é faoi \sqrt{4578} cuirtear an iolrúchán faoi \sqrt{4578} ar ceal.

$$x^{2}+\left(-\frac{\sqrt{4677521}}{\sqrt{4578}}\right)x=-\frac{20955}{\sqrt{4578}}$$

Roinn -\sqrt{4677521} faoi \sqrt{4578}.

$$x^{2}+\left(-\frac{\sqrt{21413691138}}{4578}\right)x=-\frac{20955}{\sqrt{4578}}$$

Roinn -20955 faoi \sqrt{4578}.

$$x^{2}+\left(-\frac{\sqrt{21413691138}}{4578}\right)x=-\frac{6985\sqrt{4578}}{1526}$$

Roinn -\frac{\sqrt{21413691138}}{4578}, comhéifeacht an téarma x, faoi 2 chun -\frac{\sqrt{21413691138}}{9156} a fháil. Ansin suimigh uimhir chearnach -\frac{\sqrt{21413691138}}{9156} leis an dá thaobh den chothromóid. Déanann an chéim seo slánchearnóg de thaobh clé na cothromóide.

$$x^{2}+\left(-\frac{\sqrt{21413691138}}{4578}\right)x+\left(-\frac{\sqrt{21413691138}}{9156}\right)^{2}=-\frac{6985\sqrt{4578}}{1526}+\left(-\frac{\sqrt{21413691138}}{9156}\right)^{2}$$

Cearnóg -\frac{\sqrt{21413691138}}{9156}.

$$x^{2}+\left(-\frac{\sqrt{21413691138}}{4578}\right)x+\frac{4677521}{18312}=-\frac{6985\sqrt{4578}}{1526}+\frac{4677521}{18312}$$

Fachtóirigh x^{2}+\left(-\frac{\sqrt{21413691138}}{4578}\right)x+\frac{4677521}{18312}. Go ginearálta, nuair x^{2}+bx+c cearnóg fhoirfe é, is féidir é a fhachtóiriú i gcónaí mar \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{\sqrt{21413691138}}{9156}\right)^{2}=-\frac{6985\sqrt{4578}}{1526}+\frac{4677521}{18312}$$

Tóg fréamh chearnach an dá thaobh den chothromóid.

$$\sqrt{\left(x-\frac{\sqrt{21413691138}}{9156}\right)^{2}}=\sqrt{-\frac{6985\sqrt{4578}}{1526}+\frac{4677521}{18312}}$$

Simpligh.

$$x-\frac{\sqrt{21413691138}}{9156}=\frac{i\sqrt{383727960\sqrt{4578}-21413691138}}{9156} x-\frac{\sqrt{21413691138}}{9156}=-\frac{i\sqrt{383727960\sqrt{4578}-21413691138}}{9156}$$

Cuir \frac{\sqrt{21413691138}}{9156} leis an dá thaobh den chothromóid.

$$x=\frac{\sqrt{21413691138}+i\sqrt{383727960\sqrt{4578}-21413691138}}{9156} x=\frac{-i\sqrt{383727960\sqrt{4578}-21413691138}+\sqrt{21413691138}}{9156}$$