Whakaoti i te 116=765x^2-1885x+122525

$$116 = 765 { x }^{ 2 } -1885x+122525$$

Whakaoti mō x (complex solution)

$$x=\frac{\sqrt{371018315}i}{1530}+\frac{377}{306}\approx 1.232026144+12.589435106i$$

$$x=-\frac{\sqrt{371018315}i}{1530}+\frac{377}{306}\approx 1.232026144-12.589435106i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$765x^{2}-1885x+122525=116$$

Tangohia te 116 mai i ngā taha e rua.

$$765x^{2}-1885x+122525-116=0$$

Tangohia te 116 i te 122525, ka 122409.

$$765x^{2}-1885x+122409=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 765 mō a, -1885 mō b, me 122409 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1885\right)±\sqrt{\left(-1885\right)^{2}-4\times 765\times 122409}}{2\times 765}$$

Pūrua -1885.

$$x=\frac{-\left(-1885\right)±\sqrt{3553225-4\times 765\times 122409}}{2\times 765}$$

Whakareatia -4 ki te 765.

$$x=\frac{-\left(-1885\right)±\sqrt{3553225-3060\times 122409}}{2\times 765}$$

Whakareatia -3060 ki te 122409.

$$x=\frac{-\left(-1885\right)±\sqrt{3553225-374571540}}{2\times 765}$$

Tāpiri 3553225 ki te -374571540.

$$x=\frac{-\left(-1885\right)±\sqrt{-371018315}}{2\times 765}$$

Tuhia te pūtakerua o te -371018315.

$$x=\frac{-\left(-1885\right)±\sqrt{371018315}i}{2\times 765}$$

Ko te tauaro o -1885 ko 1885.

$$x=\frac{1885±\sqrt{371018315}i}{2\times 765}$$

Whakareatia 2 ki te 765.

$$x=\frac{1885±\sqrt{371018315}i}{1530}$$

Nā, me whakaoti te whārite x=\frac{1885±\sqrt{371018315}i}{1530} ina he tāpiri te ±. Tāpiri 1885 ki te i\sqrt{371018315}.

$$x=\frac{1885+\sqrt{371018315}i}{1530}$$

Whakawehe 1885+i\sqrt{371018315} ki te 1530.

$$x=\frac{\sqrt{371018315}i}{1530}+\frac{377}{306}$$

Nā, me whakaoti te whārite x=\frac{1885±\sqrt{371018315}i}{1530} ina he tango te ±. Tango i\sqrt{371018315} mai i 1885.

$$x=\frac{-\sqrt{371018315}i+1885}{1530}$$

Whakawehe 1885-i\sqrt{371018315} ki te 1530.

$$x=-\frac{\sqrt{371018315}i}{1530}+\frac{377}{306}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{371018315}i}{1530}+\frac{377}{306} x=-\frac{\sqrt{371018315}i}{1530}+\frac{377}{306}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$765x^{2}-1885x+122525=116$$

Tangohia te 122525 mai i ngā taha e rua.

$$765x^{2}-1885x=116-122525$$

Tangohia te 122525 i te 116, ka -122409.

$$765x^{2}-1885x=-122409$$

Whakawehea ngā taha e rua ki te 765.

$$\frac{765x^{2}-1885x}{765}=-\frac{122409}{765}$$

Mā te whakawehe ki te 765 ka wetekia te whakareanga ki te 765.

$$x^{2}+\left(-\frac{1885}{765}\right)x=-\frac{122409}{765}$$

Whakahekea te hautanga \frac{-1885}{765} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 5.

$$x^{2}-\frac{377}{153}x=-\frac{122409}{765}$$

Whakahekea te hautanga \frac{-122409}{765} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 9.

$$x^{2}-\frac{377}{153}x=-\frac{13601}{85}$$

Whakawehea te -\frac{377}{153}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{377}{306}. Nā, tāpiria te pūrua o te -\frac{377}{306} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{377}{153}x+\left(-\frac{377}{306}\right)^{2}=-\frac{13601}{85}+\left(-\frac{377}{306}\right)^{2}$$

Pūruatia -\frac{377}{306} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{377}{153}x+\frac{142129}{93636}=-\frac{13601}{85}+\frac{142129}{93636}$$

Tāpiri -\frac{13601}{85} ki te \frac{142129}{93636} 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.

$$x^{2}-\frac{377}{153}x+\frac{142129}{93636}=-\frac{74203663}{468180}$$

Tauwehea x^{2}-\frac{377}{153}x+\frac{142129}{93636}. 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}.

$$\left(x-\frac{377}{306}\right)^{2}=-\frac{74203663}{468180}$$

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

$$\sqrt{\left(x-\frac{377}{306}\right)^{2}}=\sqrt{-\frac{74203663}{468180}}$$

Whakarūnātia.

$$x-\frac{377}{306}=\frac{\sqrt{371018315}i}{1530} x-\frac{377}{306}=-\frac{\sqrt{371018315}i}{1530}$$

Me tāpiri \frac{377}{306} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{371018315}i}{1530}+\frac{377}{306} x=-\frac{\sqrt{371018315}i}{1530}+\frac{377}{306}$$