Whakaoti i te 56=-32139x^2+13089x+71856

$$56 = -32139 { x }^{ 2 } +13089x+71856$$

Whakaoti mō x

$$x = \frac{\sqrt{1044626969} + 4363}{21426} \approx 1.712110963$$

$$x=\frac{4363-\sqrt{1044626969}}{21426}\approx -1.304848758$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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

$$-32139x^{2}+13089x+71856=56$$

Tangohia te 56 mai i ngā taha e rua.

$$-32139x^{2}+13089x+71856-56=0$$

Tangohia te 56 i te 71856, ka 71800.

$$-32139x^{2}+13089x+71800=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -32139 mō a, 13089 mō b, me 71800 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-13089±\sqrt{13089^{2}-4\left(-32139\right)\times 71800}}{2\left(-32139\right)}$$

Pūrua 13089.

$$x=\frac{-13089±\sqrt{171321921-4\left(-32139\right)\times 71800}}{2\left(-32139\right)}$$

Whakareatia -4 ki te -32139.

$$x=\frac{-13089±\sqrt{171321921+128556\times 71800}}{2\left(-32139\right)}$$

Whakareatia 128556 ki te 71800.

$$x=\frac{-13089±\sqrt{171321921+9230320800}}{2\left(-32139\right)}$$

Tāpiri 171321921 ki te 9230320800.

$$x=\frac{-13089±\sqrt{9401642721}}{2\left(-32139\right)}$$

Tuhia te pūtakerua o te 9401642721.

$$x=\frac{-13089±3\sqrt{1044626969}}{2\left(-32139\right)}$$

Whakareatia 2 ki te -32139.

$$x=\frac{-13089±3\sqrt{1044626969}}{-64278}$$

Nā, me whakaoti te whārite x=\frac{-13089±3\sqrt{1044626969}}{-64278} ina he tāpiri te ±. Tāpiri -13089 ki te 3\sqrt{1044626969}.

$$x=\frac{3\sqrt{1044626969}-13089}{-64278}$$

Whakawehe -13089+3\sqrt{1044626969} ki te -64278.

$$x=\frac{4363-\sqrt{1044626969}}{21426}$$

Nā, me whakaoti te whārite x=\frac{-13089±3\sqrt{1044626969}}{-64278} ina he tango te ±. Tango 3\sqrt{1044626969} mai i -13089.

$$x=\frac{-3\sqrt{1044626969}-13089}{-64278}$$

Whakawehe -13089-3\sqrt{1044626969} ki te -64278.

$$x=\frac{\sqrt{1044626969}+4363}{21426}$$

Kua oti te whārite te whakatau.

$$x=\frac{4363-\sqrt{1044626969}}{21426} x=\frac{\sqrt{1044626969}+4363}{21426}$$

Ngā Upane mō te Whakaoti i te Pūrua

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

$$-32139x^{2}+13089x+71856=56$$

Tangohia te 71856 mai i ngā taha e rua.

$$-32139x^{2}+13089x=56-71856$$

Tangohia te 71856 i te 56, ka -71800.

$$-32139x^{2}+13089x=-71800$$

Whakawehea ngā taha e rua ki te -32139.

$$\frac{-32139x^{2}+13089x}{-32139}=-\frac{71800}{-32139}$$

Mā te whakawehe ki te -32139 ka wetekia te whakareanga ki te -32139.

$$x^{2}+\frac{13089}{-32139}x=-\frac{71800}{-32139}$$

Whakahekea te hautanga \frac{13089}{-32139} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 3.

$$x^{2}-\frac{4363}{10713}x=-\frac{71800}{-32139}$$

Whakawehe -71800 ki te -32139.

$$x^{2}-\frac{4363}{10713}x=\frac{71800}{32139}$$

Whakawehea te -\frac{4363}{10713}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{4363}{21426}. Nā, tāpiria te pūrua o te -\frac{4363}{21426} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{4363}{10713}x+\left(-\frac{4363}{21426}\right)^{2}=\frac{71800}{32139}+\left(-\frac{4363}{21426}\right)^{2}$$

Pūruatia -\frac{4363}{21426} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{4363}{10713}x+\frac{19035769}{459073476}=\frac{71800}{32139}+\frac{19035769}{459073476}$$

Tāpiri \frac{71800}{32139} ki te \frac{19035769}{459073476} 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.

$$x^{2}-\frac{4363}{10713}x+\frac{19035769}{459073476}=\frac{1044626969}{459073476}$$

Tauwehea x^{2}-\frac{4363}{10713}x+\frac{19035769}{459073476}. 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}.

$$\left(x-\frac{4363}{21426}\right)^{2}=\frac{1044626969}{459073476}$$

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

$$\sqrt{\left(x-\frac{4363}{21426}\right)^{2}}=\sqrt{\frac{1044626969}{459073476}}$$

Whakarūnātia.

$$x-\frac{4363}{21426}=\frac{\sqrt{1044626969}}{21426} x-\frac{4363}{21426}=-\frac{\sqrt{1044626969}}{21426}$$

Me tāpiri \frac{4363}{21426} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{1044626969}+4363}{21426} x=\frac{4363-\sqrt{1044626969}}{21426}$$