Whakaoti i te 780x^2-28600x-38200=0

$$780 { x }^{ 2 } -28600x-38200=0$$

Whakaoti mō x

$$x=\frac{\sqrt{585715}}{39}+\frac{55}{3}\approx 37.956928062$$

$$x=-\frac{\sqrt{585715}}{39}+\frac{55}{3}\approx -1.290261396$$

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.

$$780x^{2}-28600x-38200=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 780 mō a, -28600 mō b, me -38200 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-28600\right)±\sqrt{\left(-28600\right)^{2}-4\times 780\left(-38200\right)}}{2\times 780}$$

Pūrua -28600.

$$x=\frac{-\left(-28600\right)±\sqrt{817960000-4\times 780\left(-38200\right)}}{2\times 780}$$

Whakareatia -4 ki te 780.

$$x=\frac{-\left(-28600\right)±\sqrt{817960000-3120\left(-38200\right)}}{2\times 780}$$

Whakareatia -3120 ki te -38200.

$$x=\frac{-\left(-28600\right)±\sqrt{817960000+119184000}}{2\times 780}$$

Tāpiri 817960000 ki te 119184000.

$$x=\frac{-\left(-28600\right)±\sqrt{937144000}}{2\times 780}$$

Tuhia te pūtakerua o te 937144000.

$$x=\frac{-\left(-28600\right)±40\sqrt{585715}}{2\times 780}$$

Ko te tauaro o -28600 ko 28600.

$$x=\frac{28600±40\sqrt{585715}}{2\times 780}$$

Whakareatia 2 ki te 780.

$$x=\frac{28600±40\sqrt{585715}}{1560}$$

Nā, me whakaoti te whārite x=\frac{28600±40\sqrt{585715}}{1560} ina he tāpiri te ±. Tāpiri 28600 ki te 40\sqrt{585715}.

$$x=\frac{40\sqrt{585715}+28600}{1560}$$

Whakawehe 28600+40\sqrt{585715} ki te 1560.

$$x=\frac{\sqrt{585715}}{39}+\frac{55}{3}$$

Nā, me whakaoti te whārite x=\frac{28600±40\sqrt{585715}}{1560} ina he tango te ±. Tango 40\sqrt{585715} mai i 28600.

$$x=\frac{28600-40\sqrt{585715}}{1560}$$

Whakawehe 28600-40\sqrt{585715} ki te 1560.

$$x=-\frac{\sqrt{585715}}{39}+\frac{55}{3}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{585715}}{39}+\frac{55}{3} x=-\frac{\sqrt{585715}}{39}+\frac{55}{3}$$

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.

$$780x^{2}-28600x-38200=0$$

Me tāpiri 38200 ki ngā taha e rua o te whārite.

$$780x^{2}-28600x-38200-\left(-38200\right)=-\left(-38200\right)$$

Mā te tango i te -38200 i a ia ake anō ka toe ko te 0.

$$780x^{2}-28600x=-\left(-38200\right)$$

Tango -38200 mai i 0.

$$780x^{2}-28600x=38200$$

Whakawehea ngā taha e rua ki te 780.

$$\frac{780x^{2}-28600x}{780}=\frac{38200}{780}$$

Mā te whakawehe ki te 780 ka wetekia te whakareanga ki te 780.

$$x^{2}+\left(-\frac{28600}{780}\right)x=\frac{38200}{780}$$

Whakahekea te hautanga \frac{-28600}{780} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 260.

$$x^{2}-\frac{110}{3}x=\frac{38200}{780}$$

Whakahekea te hautanga \frac{38200}{780} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 20.

$$x^{2}-\frac{110}{3}x=\frac{1910}{39}$$

Whakawehea te -\frac{110}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{55}{3}. Nā, tāpiria te pūrua o te -\frac{55}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{110}{3}x+\left(-\frac{55}{3}\right)^{2}=\frac{1910}{39}+\left(-\frac{55}{3}\right)^{2}$$

Pūruatia -\frac{55}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{110}{3}x+\frac{3025}{9}=\frac{1910}{39}+\frac{3025}{9}$$

Tāpiri \frac{1910}{39} ki te \frac{3025}{9} 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.

$$x^{2}-\frac{110}{3}x+\frac{3025}{9}=\frac{45055}{117}$$

Tauwehea x^{2}-\frac{110}{3}x+\frac{3025}{9}. 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}.

$$\left(x-\frac{55}{3}\right)^{2}=\frac{45055}{117}$$

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

$$\sqrt{\left(x-\frac{55}{3}\right)^{2}}=\sqrt{\frac{45055}{117}}$$

Whakarūnātia.

$$x-\frac{55}{3}=\frac{\sqrt{585715}}{39} x-\frac{55}{3}=-\frac{\sqrt{585715}}{39}$$

Me tāpiri \frac{55}{3} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{585715}}{39}+\frac{55}{3} x=-\frac{\sqrt{585715}}{39}+\frac{55}{3}$$