Whakaoti i te 1530quadx^2-30x-470=0

$$1530 \quad x ^ { 2 } - 30 x - 470 = 0$$

Whakaoti mō x

$$x=\frac{\sqrt{3197}+1}{102}\approx 0.564137449$$

$$x=\frac{1-\sqrt{3197}}{102}\approx -0.544529606$$

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.

$$1530x^{2}-30x-470=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1530 mō a, -30 mō b, me -470 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-30\right)±\sqrt{\left(-30\right)^{2}-4\times 1530\left(-470\right)}}{2\times 1530}$$

Pūrua -30.

$$x=\frac{-\left(-30\right)±\sqrt{900-4\times 1530\left(-470\right)}}{2\times 1530}$$

Whakareatia -4 ki te 1530.

$$x=\frac{-\left(-30\right)±\sqrt{900-6120\left(-470\right)}}{2\times 1530}$$

Whakareatia -6120 ki te -470.

$$x=\frac{-\left(-30\right)±\sqrt{900+2876400}}{2\times 1530}$$

Tāpiri 900 ki te 2876400.

$$x=\frac{-\left(-30\right)±\sqrt{2877300}}{2\times 1530}$$

Tuhia te pūtakerua o te 2877300.

$$x=\frac{-\left(-30\right)±30\sqrt{3197}}{2\times 1530}$$

Ko te tauaro o -30 ko 30.

$$x=\frac{30±30\sqrt{3197}}{2\times 1530}$$

Whakareatia 2 ki te 1530.

$$x=\frac{30±30\sqrt{3197}}{3060}$$

Nā, me whakaoti te whārite x=\frac{30±30\sqrt{3197}}{3060} ina he tāpiri te ±. Tāpiri 30 ki te 30\sqrt{3197}.

$$x=\frac{30\sqrt{3197}+30}{3060}$$

Whakawehe 30+30\sqrt{3197} ki te 3060.

$$x=\frac{\sqrt{3197}+1}{102}$$

Nā, me whakaoti te whārite x=\frac{30±30\sqrt{3197}}{3060} ina he tango te ±. Tango 30\sqrt{3197} mai i 30.

$$x=\frac{30-30\sqrt{3197}}{3060}$$

Whakawehe 30-30\sqrt{3197} ki te 3060.

$$x=\frac{1-\sqrt{3197}}{102}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{3197}+1}{102} x=\frac{1-\sqrt{3197}}{102}$$

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.

$$1530x^{2}-30x-470=0$$

Me tāpiri 470 ki ngā taha e rua o te whārite.

$$1530x^{2}-30x-470-\left(-470\right)=-\left(-470\right)$$

Mā te tango i te -470 i a ia ake anō ka toe ko te 0.

$$1530x^{2}-30x=-\left(-470\right)$$

Tango -470 mai i 0.

$$1530x^{2}-30x=470$$

Whakawehea ngā taha e rua ki te 1530.

$$\frac{1530x^{2}-30x}{1530}=\frac{470}{1530}$$

Mā te whakawehe ki te 1530 ka wetekia te whakareanga ki te 1530.

$$x^{2}+\left(-\frac{30}{1530}\right)x=\frac{470}{1530}$$

Whakahekea te hautanga \frac{-30}{1530} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 30.

$$x^{2}-\frac{1}{51}x=\frac{470}{1530}$$

Whakahekea te hautanga \frac{470}{1530} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 10.

$$x^{2}-\frac{1}{51}x=\frac{47}{153}$$

Whakawehea te -\frac{1}{51}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{102}. Nā, tāpiria te pūrua o te -\frac{1}{102} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{1}{51}x+\left(-\frac{1}{102}\right)^{2}=\frac{47}{153}+\left(-\frac{1}{102}\right)^{2}$$

Pūruatia -\frac{1}{102} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{1}{51}x+\frac{1}{10404}=\frac{47}{153}+\frac{1}{10404}$$

Tāpiri \frac{47}{153} ki te \frac{1}{10404} 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.

$$x^{2}-\frac{1}{51}x+\frac{1}{10404}=\frac{3197}{10404}$$

Tauwehea x^{2}-\frac{1}{51}x+\frac{1}{10404}. 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}.

$$\left(x-\frac{1}{102}\right)^{2}=\frac{3197}{10404}$$

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

$$\sqrt{\left(x-\frac{1}{102}\right)^{2}}=\sqrt{\frac{3197}{10404}}$$

Whakarūnātia.

$$x-\frac{1}{102}=\frac{\sqrt{3197}}{102} x-\frac{1}{102}=-\frac{\sqrt{3197}}{102}$$

Me tāpiri \frac{1}{102} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{3197}+1}{102} x=\frac{1-\sqrt{3197}}{102}$$