Whakaoti i te sqrt{4578}x^2-sqrt{4677521}x+31478=10523

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

Whakaoti mō x (complex solution)

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

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

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 10523 mai i ngā taha e rua.

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

Tangohia te 10523 i te 31478, ka 20955.

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

Ko ngā whārite katoa o te āhua ax^{2}+bx+c=0 ka taea te whakaoti mā te whakamahi i te tikanga tātai pūrua: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. E rua ngā otinga ka puta i te tikanga tātai pūrua, ko tētahi ina he tāpiri a ±, ā, ko tētahi ina he tango.

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

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi \sqrt{4578} mō a, -\sqrt{4677521} mō b, me 20955 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Pūrua -\sqrt{4677521}.

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

Whakareatia -4 ki te \sqrt{4578}.

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

Whakareatia -4\sqrt{4578} ki te 20955.

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

Tuhia te pūtakerua o te 4677521-83820\sqrt{4578}.

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

Ko te tauaro o -\sqrt{4677521} ko \sqrt{4677521}.

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

Nā, me whakaoti te whārite x=\frac{\sqrt{4677521}±i\sqrt{-\left(4677521-83820\sqrt{4578}\right)}}{2\sqrt{4578}} ina he tāpiri te ±. Tāpiri \sqrt{4677521} ki te i\sqrt{-\left(4677521-83820\sqrt{4578}\right)}.

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

Whakawehe \sqrt{4677521}+i\sqrt{-4677521+83820\sqrt{4578}} ki te 2\sqrt{4578}.

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

Nā, me whakaoti te whārite x=\frac{\sqrt{4677521}±i\sqrt{-\left(4677521-83820\sqrt{4578}\right)}}{2\sqrt{4578}} ina he tango te ±. Tango i\sqrt{-\left(4677521-83820\sqrt{4578}\right)} mai i \sqrt{4677521}.

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

Whakawehe \sqrt{4677521}-i\sqrt{-4677521+83820\sqrt{4578}} ki te 2\sqrt{4578}.

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

Kua oti te whārite te whakatau.

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

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te 31478 mai i ngā taha e rua.

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

Tangohia te 31478 i te 10523, ka -20955.

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

Ko ngā whārite pūrua pēnei i tēnei nā ka taea te whakaoti mā te whakaoti i te pūrua. Hei whakaoti i te pūrua, ko te whārite me mātua tuhi ki te āhua x^{2}+bx=c.

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

Whakawehea ngā taha e rua ki te \sqrt{4578}.

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

Mā te whakawehe ki te \sqrt{4578} ka wetekia te whakareanga ki te \sqrt{4578}.

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

Whakawehe -\sqrt{4677521} ki te \sqrt{4578}.

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

Whakawehe -20955 ki te \sqrt{4578}.

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

Whakawehea te -\frac{\sqrt{21413691138}}{4578}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{\sqrt{21413691138}}{9156}. Nā, tāpiria te pūrua o te -\frac{\sqrt{21413691138}}{9156} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

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

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

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

Tauwehea x^{2}+\left(-\frac{\sqrt{21413691138}}{4578}\right)x+\frac{4677521}{18312}. Ko te tikanga pūnoa, ina ko x^{2}+bx+c he pūrua tika pūrua tika pū, ka taea taua mea te tauwehea i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

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

Tuhia te pūtakerua o ngā taha e rua o te whārite.

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

Whakarūnātia.

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

Me tāpiri \frac{\sqrt{21413691138}}{9156} ki ngā taha e rua o te whārite.

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