Whakaoti i te 80x^2-100x+32=0

$$80 x ^ { 2 } - 100 x + 32 = 0$$

Whakaoti mō x (complex solution)

$$x=\frac{\sqrt{15}i}{40}+\frac{5}{8}\approx 0.625+0.096824584i$$

$$x=-\frac{\sqrt{15}i}{40}+\frac{5}{8}\approx 0.625-0.096824584i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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.

$$80x^{2}-100x+32=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 80 mō a, -100 mō b, me 32 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-100\right)±\sqrt{\left(-100\right)^{2}-4\times 80\times 32}}{2\times 80}$$

Pūrua -100.

$$x=\frac{-\left(-100\right)±\sqrt{10000-4\times 80\times 32}}{2\times 80}$$

Whakareatia -4 ki te 80.

$$x=\frac{-\left(-100\right)±\sqrt{10000-320\times 32}}{2\times 80}$$

Whakareatia -320 ki te 32.

$$x=\frac{-\left(-100\right)±\sqrt{10000-10240}}{2\times 80}$$

Tāpiri 10000 ki te -10240.

$$x=\frac{-\left(-100\right)±\sqrt{-240}}{2\times 80}$$

Tuhia te pūtakerua o te -240.

$$x=\frac{-\left(-100\right)±4\sqrt{15}i}{2\times 80}$$

Ko te tauaro o -100 ko 100.

$$x=\frac{100±4\sqrt{15}i}{2\times 80}$$

Whakareatia 2 ki te 80.

$$x=\frac{100±4\sqrt{15}i}{160}$$

Nā, me whakaoti te whārite x=\frac{100±4\sqrt{15}i}{160} ina he tāpiri te ±. Tāpiri 100 ki te 4i\sqrt{15}.

$$x=\frac{100+4\sqrt{15}i}{160}$$

Whakawehe 100+4i\sqrt{15} ki te 160.

$$x=\frac{\sqrt{15}i}{40}+\frac{5}{8}$$

Nā, me whakaoti te whārite x=\frac{100±4\sqrt{15}i}{160} ina he tango te ±. Tango 4i\sqrt{15} mai i 100.

$$x=\frac{-4\sqrt{15}i+100}{160}$$

Whakawehe 100-4i\sqrt{15} ki te 160.

$$x=-\frac{\sqrt{15}i}{40}+\frac{5}{8}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{15}i}{40}+\frac{5}{8} x=-\frac{\sqrt{15}i}{40}+\frac{5}{8}$$

Ngā Upane mō te Whakaoti i te Pūrua

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.

$$80x^{2}-100x+32=0$$

Me tango 32 mai i ngā taha e rua o te whārite.

$$80x^{2}-100x+32-32=-32$$

Mā te tango i te 32 i a ia ake anō ka toe ko te 0.

$$80x^{2}-100x=-32$$

Whakawehea ngā taha e rua ki te 80.

$$\frac{80x^{2}-100x}{80}=-\frac{32}{80}$$

Mā te whakawehe ki te 80 ka wetekia te whakareanga ki te 80.

$$x^{2}+\left(-\frac{100}{80}\right)x=-\frac{32}{80}$$

Whakahekea te hautanga \frac{-100}{80} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 20.

$$x^{2}-\frac{5}{4}x=-\frac{32}{80}$$

Whakahekea te hautanga \frac{-32}{80} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 16.

$$x^{2}-\frac{5}{4}x=-\frac{2}{5}$$

Whakawehea te -\frac{5}{4}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{5}{8}. Nā, tāpiria te pūrua o te -\frac{5}{8} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{5}{4}x+\left(-\frac{5}{8}\right)^{2}=-\frac{2}{5}+\left(-\frac{5}{8}\right)^{2}$$

Pūruatia -\frac{5}{8} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{5}{4}x+\frac{25}{64}=-\frac{2}{5}+\frac{25}{64}$$

Tāpiri -\frac{2}{5} ki te \frac{25}{64} 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.

$$x^{2}-\frac{5}{4}x+\frac{25}{64}=-\frac{3}{320}$$

Tauwehea x^{2}-\frac{5}{4}x+\frac{25}{64}. 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}.

$$\left(x-\frac{5}{8}\right)^{2}=-\frac{3}{320}$$

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

$$\sqrt{\left(x-\frac{5}{8}\right)^{2}}=\sqrt{-\frac{3}{320}}$$

Whakarūnātia.

$$x-\frac{5}{8}=\frac{\sqrt{15}i}{40} x-\frac{5}{8}=-\frac{\sqrt{15}i}{40}$$

Me tāpiri \frac{5}{8} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{15}i}{40}+\frac{5}{8} x=-\frac{\sqrt{15}i}{40}+\frac{5}{8}$$