Whakaoti i te 43897+204x^2=-59414x^2+13216x+52929

$$43897+204 { x }^{ 2 } =-59414 { x }^{ 2 } +13216x+52929$$

Whakaoti mō x

$$x=\frac{2\sqrt{36383465}+3304}{29809}\approx 0.515540325$$

$$x=\frac{3304-2\sqrt{36383465}}{29809}\approx -0.293862308$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Me tāpiri te 59414x^{2} ki ngā taha e rua.

$$43897+204x^{2}+59414x^{2}=13216x+52929$$

Pahekotia te 204x^{2} me 59414x^{2}, ka 59618x^{2}.

$$43897+59618x^{2}=13216x+52929$$

Tangohia te 13216x mai i ngā taha e rua.

$$43897+59618x^{2}-13216x=52929$$

Tangohia te 52929 mai i ngā taha e rua.

$$43897+59618x^{2}-13216x-52929=0$$

Tangohia te 52929 i te 43897, ka -9032.

$$-9032+59618x^{2}-13216x=0$$

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.

$$59618x^{2}-13216x-9032=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 59618 mō a, -13216 mō b, me -9032 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-13216\right)±\sqrt{\left(-13216\right)^{2}-4\times 59618\left(-9032\right)}}{2\times 59618}$$

Pūrua -13216.

$$x=\frac{-\left(-13216\right)±\sqrt{174662656-4\times 59618\left(-9032\right)}}{2\times 59618}$$

Whakareatia -4 ki te 59618.

$$x=\frac{-\left(-13216\right)±\sqrt{174662656-238472\left(-9032\right)}}{2\times 59618}$$

Whakareatia -238472 ki te -9032.

$$x=\frac{-\left(-13216\right)±\sqrt{174662656+2153879104}}{2\times 59618}$$

Tāpiri 174662656 ki te 2153879104.

$$x=\frac{-\left(-13216\right)±\sqrt{2328541760}}{2\times 59618}$$

Tuhia te pūtakerua o te 2328541760.

$$x=\frac{-\left(-13216\right)±8\sqrt{36383465}}{2\times 59618}$$

Ko te tauaro o -13216 ko 13216.

$$x=\frac{13216±8\sqrt{36383465}}{2\times 59618}$$

Whakareatia 2 ki te 59618.

$$x=\frac{13216±8\sqrt{36383465}}{119236}$$

Nā, me whakaoti te whārite x=\frac{13216±8\sqrt{36383465}}{119236} ina he tāpiri te ±. Tāpiri 13216 ki te 8\sqrt{36383465}.

$$x=\frac{8\sqrt{36383465}+13216}{119236}$$

Whakawehe 13216+8\sqrt{36383465} ki te 119236.

$$x=\frac{2\sqrt{36383465}+3304}{29809}$$

Nā, me whakaoti te whārite x=\frac{13216±8\sqrt{36383465}}{119236} ina he tango te ±. Tango 8\sqrt{36383465} mai i 13216.

$$x=\frac{13216-8\sqrt{36383465}}{119236}$$

Whakawehe 13216-8\sqrt{36383465} ki te 119236.

$$x=\frac{3304-2\sqrt{36383465}}{29809}$$

Kua oti te whārite te whakatau.

$$x=\frac{2\sqrt{36383465}+3304}{29809} x=\frac{3304-2\sqrt{36383465}}{29809}$$

Ngā Upane mō te Whakaoti i te Pūrua

Me tāpiri te 59414x^{2} ki ngā taha e rua.

$$43897+204x^{2}+59414x^{2}=13216x+52929$$

Pahekotia te 204x^{2} me 59414x^{2}, ka 59618x^{2}.

$$43897+59618x^{2}=13216x+52929$$

Tangohia te 13216x mai i ngā taha e rua.

$$43897+59618x^{2}-13216x=52929$$

Tangohia te 43897 mai i ngā taha e rua.

$$59618x^{2}-13216x=52929-43897$$

Tangohia te 43897 i te 52929, ka 9032.

$$59618x^{2}-13216x=9032$$

Whakawehea ngā taha e rua ki te 59618.

$$\frac{59618x^{2}-13216x}{59618}=\frac{9032}{59618}$$

Mā te whakawehe ki te 59618 ka wetekia te whakareanga ki te 59618.

$$x^{2}+\left(-\frac{13216}{59618}\right)x=\frac{9032}{59618}$$

Whakahekea te hautanga \frac{-13216}{59618} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}-\frac{6608}{29809}x=\frac{9032}{59618}$$

Whakahekea te hautanga \frac{9032}{59618} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}-\frac{6608}{29809}x=\frac{4516}{29809}$$

Whakawehea te -\frac{6608}{29809}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{3304}{29809}. Nā, tāpiria te pūrua o te -\frac{3304}{29809} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{6608}{29809}x+\left(-\frac{3304}{29809}\right)^{2}=\frac{4516}{29809}+\left(-\frac{3304}{29809}\right)^{2}$$

Pūruatia -\frac{3304}{29809} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{6608}{29809}x+\frac{10916416}{888576481}=\frac{4516}{29809}+\frac{10916416}{888576481}$$

Tāpiri \frac{4516}{29809} ki te \frac{10916416}{888576481} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x^{2}-\frac{6608}{29809}x+\frac{10916416}{888576481}=\frac{145533860}{888576481}$$

Tauwehea x^{2}-\frac{6608}{29809}x+\frac{10916416}{888576481}. 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}.

$$\left(x-\frac{3304}{29809}\right)^{2}=\frac{145533860}{888576481}$$

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

$$\sqrt{\left(x-\frac{3304}{29809}\right)^{2}}=\sqrt{\frac{145533860}{888576481}}$$

Whakarūnātia.

$$x-\frac{3304}{29809}=\frac{2\sqrt{36383465}}{29809} x-\frac{3304}{29809}=-\frac{2\sqrt{36383465}}{29809}$$

Me tāpiri \frac{3304}{29809} ki ngā taha e rua o te whārite.

$$x=\frac{2\sqrt{36383465}+3304}{29809} x=\frac{3304-2\sqrt{36383465}}{29809}$$