Whakaoti i te 0.0015x^2-0.14x-30=0

$$0.0015 { x }^{ 2 } -0.14x-30 = 0$$

Whakaoti mō x

$$x = \frac{20 \sqrt{499} + 140}{3} \approx 195.588719358$$

$$x=\frac{140-20\sqrt{499}}{3}\approx -102.255386025$$

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.

$$0.0015x^{2}-0.14x-30=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 0.0015 mō a, -0.14 mō b, me -30 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-0.14\right)±\sqrt{\left(-0.14\right)^{2}-4\times 0.0015\left(-30\right)}}{2\times 0.0015}$$

Pūruatia -0.14 mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x=\frac{-\left(-0.14\right)±\sqrt{0.0196-4\times 0.0015\left(-30\right)}}{2\times 0.0015}$$

Whakareatia -4 ki te 0.0015.

$$x=\frac{-\left(-0.14\right)±\sqrt{0.0196-0.006\left(-30\right)}}{2\times 0.0015}$$

Whakareatia -0.006 ki te -30.

$$x=\frac{-\left(-0.14\right)±\sqrt{0.0196+0.18}}{2\times 0.0015}$$

Tāpiri 0.0196 ki te 0.18 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.

$$x=\frac{-\left(-0.14\right)±\sqrt{0.1996}}{2\times 0.0015}$$

Tuhia te pūtakerua o te 0.1996.

$$x=\frac{-\left(-0.14\right)±\frac{\sqrt{499}}{50}}{2\times 0.0015}$$

Ko te tauaro o -0.14 ko 0.14.

$$x=\frac{0.14±\frac{\sqrt{499}}{50}}{2\times 0.0015}$$

Whakareatia 2 ki te 0.0015.

$$x=\frac{0.14±\frac{\sqrt{499}}{50}}{0.003}$$

Nā, me whakaoti te whārite x=\frac{0.14±\frac{\sqrt{499}}{50}}{0.003} ina he tāpiri te ±. Tāpiri 0.14 ki te \frac{\sqrt{499}}{50}.

$$x=\frac{\sqrt{499}+7}{0.003\times 50}$$

Whakawehe \frac{7+\sqrt{499}}{50} ki te 0.003 mā te whakarea \frac{7+\sqrt{499}}{50} ki te tau huripoki o 0.003.

$$x=\frac{20\sqrt{499}+140}{3}$$

Nā, me whakaoti te whārite x=\frac{0.14±\frac{\sqrt{499}}{50}}{0.003} ina he tango te ±. Tango \frac{\sqrt{499}}{50} mai i 0.14.

$$x=\frac{7-\sqrt{499}}{0.003\times 50}$$

Whakawehe \frac{7-\sqrt{499}}{50} ki te 0.003 mā te whakarea \frac{7-\sqrt{499}}{50} ki te tau huripoki o 0.003.

$$x=\frac{140-20\sqrt{499}}{3}$$

Kua oti te whārite te whakatau.

$$x=\frac{20\sqrt{499}+140}{3} x=\frac{140-20\sqrt{499}}{3}$$

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.

$$0.0015x^{2}-0.14x-30=0$$

Me tāpiri 30 ki ngā taha e rua o te whārite.

$$0.0015x^{2}-0.14x-30-\left(-30\right)=-\left(-30\right)$$

Mā te tango i te -30 i a ia ake anō ka toe ko te 0.

$$0.0015x^{2}-0.14x=-\left(-30\right)$$

Tango -30 mai i 0.

$$0.0015x^{2}-0.14x=30$$

Whakawehea ngā taha e rua o te whārite ki te 0.0015, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$\frac{0.0015x^{2}-0.14x}{0.0015}=\frac{30}{0.0015}$$

Mā te whakawehe ki te 0.0015 ka wetekia te whakareanga ki te 0.0015.

$$x^{2}+\left(-\frac{0.14}{0.0015}\right)x=\frac{30}{0.0015}$$

Whakawehe -0.14 ki te 0.0015 mā te whakarea -0.14 ki te tau huripoki o 0.0015.

$$x^{2}-\frac{280}{3}x=\frac{30}{0.0015}$$

Whakawehe 30 ki te 0.0015 mā te whakarea 30 ki te tau huripoki o 0.0015.

$$x^{2}-\frac{280}{3}x=20000$$

Whakawehea te -\frac{280}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{140}{3}. Nā, tāpiria te pūrua o te -\frac{140}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{280}{3}x+\left(-\frac{140}{3}\right)^{2}=20000+\left(-\frac{140}{3}\right)^{2}$$

Pūruatia -\frac{140}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{280}{3}x+\frac{19600}{9}=20000+\frac{19600}{9}$$

Tāpiri 20000 ki te \frac{19600}{9}.

$$x^{2}-\frac{280}{3}x+\frac{19600}{9}=\frac{199600}{9}$$

Tauwehea x^{2}-\frac{280}{3}x+\frac{19600}{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}.

$$\left(x-\frac{140}{3}\right)^{2}=\frac{199600}{9}$$

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

$$\sqrt{\left(x-\frac{140}{3}\right)^{2}}=\sqrt{\frac{199600}{9}}$$

Whakarūnātia.

$$x-\frac{140}{3}=\frac{20\sqrt{499}}{3} x-\frac{140}{3}=-\frac{20\sqrt{499}}{3}$$

Me tāpiri \frac{140}{3} ki ngā taha e rua o te whārite.

$$x=\frac{20\sqrt{499}+140}{3} x=\frac{140-20\sqrt{499}}{3}$$