Whakaoti i te -3x^2+16x+128=0

$$-3 { x }^{ 2 } +16x+128=0$$

Whakaoti mō x

$$x = \frac{8 \sqrt{7} + 8}{3} \approx 9.722003496$$

$$x=\frac{8-8\sqrt{7}}{3}\approx -4.388670163$$

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.

$$-3x^{2}+16x+128=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -3 mō a, 16 mō b, me 128 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-16±\sqrt{16^{2}-4\left(-3\right)\times 128}}{2\left(-3\right)}$$

Pūrua 16.

$$x=\frac{-16±\sqrt{256-4\left(-3\right)\times 128}}{2\left(-3\right)}$$

Whakareatia -4 ki te -3.

$$x=\frac{-16±\sqrt{256+12\times 128}}{2\left(-3\right)}$$

Whakareatia 12 ki te 128.

$$x=\frac{-16±\sqrt{256+1536}}{2\left(-3\right)}$$

Tāpiri 256 ki te 1536.

$$x=\frac{-16±\sqrt{1792}}{2\left(-3\right)}$$

Tuhia te pūtakerua o te 1792.

$$x=\frac{-16±16\sqrt{7}}{2\left(-3\right)}$$

Whakareatia 2 ki te -3.

$$x=\frac{-16±16\sqrt{7}}{-6}$$

Nā, me whakaoti te whārite x=\frac{-16±16\sqrt{7}}{-6} ina he tāpiri te ±. Tāpiri -16 ki te 16\sqrt{7}.

$$x=\frac{16\sqrt{7}-16}{-6}$$

Whakawehe -16+16\sqrt{7} ki te -6.

$$x=\frac{8-8\sqrt{7}}{3}$$

Nā, me whakaoti te whārite x=\frac{-16±16\sqrt{7}}{-6} ina he tango te ±. Tango 16\sqrt{7} mai i -16.

$$x=\frac{-16\sqrt{7}-16}{-6}$$

Whakawehe -16-16\sqrt{7} ki te -6.

$$x=\frac{8\sqrt{7}+8}{3}$$

Kua oti te whārite te whakatau.

$$x=\frac{8-8\sqrt{7}}{3} x=\frac{8\sqrt{7}+8}{3}$$

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.

$$-3x^{2}+16x+128=0$$

Me tango 128 mai i ngā taha e rua o te whārite.

$$-3x^{2}+16x+128-128=-128$$

Mā te tango i te 128 i a ia ake anō ka toe ko te 0.

$$-3x^{2}+16x=-128$$

Whakawehea ngā taha e rua ki te -3.

$$\frac{-3x^{2}+16x}{-3}=-\frac{128}{-3}$$

Mā te whakawehe ki te -3 ka wetekia te whakareanga ki te -3.

$$x^{2}+\frac{16}{-3}x=-\frac{128}{-3}$$

Whakawehe 16 ki te -3.

$$x^{2}-\frac{16}{3}x=-\frac{128}{-3}$$

Whakawehe -128 ki te -3.

$$x^{2}-\frac{16}{3}x=\frac{128}{3}$$

Whakawehea te -\frac{16}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{8}{3}. Nā, tāpiria te pūrua o te -\frac{8}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{16}{3}x+\left(-\frac{8}{3}\right)^{2}=\frac{128}{3}+\left(-\frac{8}{3}\right)^{2}$$

Pūruatia -\frac{8}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{16}{3}x+\frac{64}{9}=\frac{128}{3}+\frac{64}{9}$$

Tāpiri \frac{128}{3} ki te \frac{64}{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.

$$x^{2}-\frac{16}{3}x+\frac{64}{9}=\frac{448}{9}$$

Tauwehea x^{2}-\frac{16}{3}x+\frac{64}{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}.

$$\left(x-\frac{8}{3}\right)^{2}=\frac{448}{9}$$

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

$$\sqrt{\left(x-\frac{8}{3}\right)^{2}}=\sqrt{\frac{448}{9}}$$

Whakarūnātia.

$$x-\frac{8}{3}=\frac{8\sqrt{7}}{3} x-\frac{8}{3}=-\frac{8\sqrt{7}}{3}$$

Me tāpiri \frac{8}{3} ki ngā taha e rua o te whārite.

$$x=\frac{8\sqrt{7}+8}{3} x=\frac{8-8\sqrt{7}}{3}$$