Whakaoti i te 8x^2-x-180=0

$$8 x ^ { 2 } - x - 180 = 0$$

Whakaoti mō x

$$x = \frac{\sqrt{5761} + 1}{16} \approx 4.806328227$$

$$x=\frac{1-\sqrt{5761}}{16}\approx -4.681328227$$

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.

$$8x^{2}-x-180=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 8 mō a, -1 mō b, me -180 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times 8\left(-180\right)}}{2\times 8}$$

Whakareatia -4 ki te 8.

$$x=\frac{-\left(-1\right)±\sqrt{1-32\left(-180\right)}}{2\times 8}$$

Whakareatia -32 ki te -180.

$$x=\frac{-\left(-1\right)±\sqrt{1+5760}}{2\times 8}$$

Tāpiri 1 ki te 5760.

$$x=\frac{-\left(-1\right)±\sqrt{5761}}{2\times 8}$$

Ko te tauaro o -1 ko 1.

$$x=\frac{1±\sqrt{5761}}{2\times 8}$$

Whakareatia 2 ki te 8.

$$x=\frac{1±\sqrt{5761}}{16}$$

Nā, me whakaoti te whārite x=\frac{1±\sqrt{5761}}{16} ina he tāpiri te ±. Tāpiri 1 ki te \sqrt{5761}.

$$x=\frac{\sqrt{5761}+1}{16}$$

Nā, me whakaoti te whārite x=\frac{1±\sqrt{5761}}{16} ina he tango te ±. Tango \sqrt{5761} mai i 1.

$$x=\frac{1-\sqrt{5761}}{16}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{5761}+1}{16} x=\frac{1-\sqrt{5761}}{16}$$

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.

$$8x^{2}-x-180=0$$

Me tāpiri 180 ki ngā taha e rua o te whārite.

$$8x^{2}-x-180-\left(-180\right)=-\left(-180\right)$$

Mā te tango i te -180 i a ia ake anō ka toe ko te 0.

$$8x^{2}-x=-\left(-180\right)$$

Tango -180 mai i 0.

$$8x^{2}-x=180$$

Whakawehea ngā taha e rua ki te 8.

$$\frac{8x^{2}-x}{8}=\frac{180}{8}$$

Mā te whakawehe ki te 8 ka wetekia te whakareanga ki te 8.

$$x^{2}-\frac{1}{8}x=\frac{180}{8}$$

Whakahekea te hautanga \frac{180}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x^{2}-\frac{1}{8}x=\frac{45}{2}$$

Whakawehea te -\frac{1}{8}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{16}. Nā, tāpiria te pūrua o te -\frac{1}{16} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{1}{8}x+\left(-\frac{1}{16}\right)^{2}=\frac{45}{2}+\left(-\frac{1}{16}\right)^{2}$$

Pūruatia -\frac{1}{16} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{1}{8}x+\frac{1}{256}=\frac{45}{2}+\frac{1}{256}$$

Tāpiri \frac{45}{2} ki te \frac{1}{256} 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.

$$x^{2}-\frac{1}{8}x+\frac{1}{256}=\frac{5761}{256}$$

Tauwehea x^{2}-\frac{1}{8}x+\frac{1}{256}. 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}.

$$\left(x-\frac{1}{16}\right)^{2}=\frac{5761}{256}$$

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

$$\sqrt{\left(x-\frac{1}{16}\right)^{2}}=\sqrt{\frac{5761}{256}}$$

Whakarūnātia.

$$x-\frac{1}{16}=\frac{\sqrt{5761}}{16} x-\frac{1}{16}=-\frac{\sqrt{5761}}{16}$$

Me tāpiri \frac{1}{16} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{5761}+1}{16} x=\frac{1-\sqrt{5761}}{16}$$