Whakaoti i te 8.314x+0.0149x^2-1000=0

$$8.314x+0.0149 { x }^{ 2 } -1000 = 0$$

Whakaoti mō x

$$x = \frac{10 \sqrt{32180649} - 41570}{149} \approx 101.731479501$$

$$x=\frac{-10\sqrt{32180649}-41570}{149}\approx -659.718056682$$

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.

$$0.0149x^{2}+8.314x-1000=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 0.0149 mō a, 8.314 mō b, me -1000 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8.314±\sqrt{8.314^{2}-4\times 0.0149\left(-1000\right)}}{2\times 0.0149}$$

Pūruatia 8.314 mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x=\frac{-8.314±\sqrt{69.122596-4\times 0.0149\left(-1000\right)}}{2\times 0.0149}$$

Whakareatia -4 ki te 0.0149.

$$x=\frac{-8.314±\sqrt{69.122596-0.0596\left(-1000\right)}}{2\times 0.0149}$$

Whakareatia -0.0596 ki te -1000.

$$x=\frac{-8.314±\sqrt{69.122596+59.6}}{2\times 0.0149}$$

Tāpiri 69.122596 ki te 59.6 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.

$$x=\frac{-8.314±\sqrt{128.722596}}{2\times 0.0149}$$

Tuhia te pūtakerua o te 128.722596.

$$x=\frac{-8.314±\frac{\sqrt{32180649}}{500}}{2\times 0.0149}$$

Whakareatia 2 ki te 0.0149.

$$x=\frac{-8.314±\frac{\sqrt{32180649}}{500}}{0.0298}$$

Nā, me whakaoti te whārite x=\frac{-8.314±\frac{\sqrt{32180649}}{500}}{0.0298} ina he tāpiri te ±. Tāpiri -8.314 ki te \frac{\sqrt{32180649}}{500}.

$$x=\frac{\sqrt{32180649}-4157}{0.0298\times 500}$$

Whakawehe \frac{-4157+\sqrt{32180649}}{500} ki te 0.0298 mā te whakarea \frac{-4157+\sqrt{32180649}}{500} ki te tau huripoki o 0.0298.

$$x=\frac{10\sqrt{32180649}-41570}{149}$$

Nā, me whakaoti te whārite x=\frac{-8.314±\frac{\sqrt{32180649}}{500}}{0.0298} ina he tango te ±. Tango \frac{\sqrt{32180649}}{500} mai i -8.314.

$$x=\frac{-\sqrt{32180649}-4157}{0.0298\times 500}$$

Whakawehe \frac{-4157-\sqrt{32180649}}{500} ki te 0.0298 mā te whakarea \frac{-4157-\sqrt{32180649}}{500} ki te tau huripoki o 0.0298.

$$x=\frac{-10\sqrt{32180649}-41570}{149}$$

Kua oti te whārite te whakatau.

$$x=\frac{10\sqrt{32180649}-41570}{149} x=\frac{-10\sqrt{32180649}-41570}{149}$$

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.

$$0.0149x^{2}+8.314x-1000=0$$

Me tāpiri 1000 ki ngā taha e rua o te whārite.

$$0.0149x^{2}+8.314x-1000-\left(-1000\right)=-\left(-1000\right)$$

Mā te tango i te -1000 i a ia ake anō ka toe ko te 0.

$$0.0149x^{2}+8.314x=-\left(-1000\right)$$

Tango -1000 mai i 0.

$$0.0149x^{2}+8.314x=1000$$

Whakawehea ngā taha e rua o te whārite ki te 0.0149, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$\frac{0.0149x^{2}+8.314x}{0.0149}=\frac{1000}{0.0149}$$

Mā te whakawehe ki te 0.0149 ka wetekia te whakareanga ki te 0.0149.

$$x^{2}+\frac{8.314}{0.0149}x=\frac{1000}{0.0149}$$

Whakawehe 8.314 ki te 0.0149 mā te whakarea 8.314 ki te tau huripoki o 0.0149.

$$x^{2}+\frac{83140}{149}x=\frac{1000}{0.0149}$$

Whakawehe 1000 ki te 0.0149 mā te whakarea 1000 ki te tau huripoki o 0.0149.

$$x^{2}+\frac{83140}{149}x=\frac{10000000}{149}$$

Whakawehea te \frac{83140}{149}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{41570}{149}. Nā, tāpiria te pūrua o te \frac{41570}{149} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{83140}{149}x+\frac{41570}{149}^{2}=\frac{10000000}{149}+\frac{41570}{149}^{2}$$

Pūruatia \frac{41570}{149} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{83140}{149}x+\frac{1728064900}{22201}=\frac{10000000}{149}+\frac{1728064900}{22201}$$

Tāpiri \frac{10000000}{149} ki te \frac{1728064900}{22201} 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.

$$x^{2}+\frac{83140}{149}x+\frac{1728064900}{22201}=\frac{3218064900}{22201}$$

Tauwehea x^{2}+\frac{83140}{149}x+\frac{1728064900}{22201}. 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}.

$$\left(x+\frac{41570}{149}\right)^{2}=\frac{3218064900}{22201}$$

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

$$\sqrt{\left(x+\frac{41570}{149}\right)^{2}}=\sqrt{\frac{3218064900}{22201}}$$

Whakarūnātia.

$$x+\frac{41570}{149}=\frac{10\sqrt{32180649}}{149} x+\frac{41570}{149}=-\frac{10\sqrt{32180649}}{149}$$

Me tango \frac{41570}{149} mai i ngā taha e rua o te whārite.

$$x=\frac{10\sqrt{32180649}-41570}{149} x=\frac{-10\sqrt{32180649}-41570}{149}$$