Whakaoti i te 5x^2=3x-7

$$5 x ^ { 2 } = 3 x - 7$$

Whakaoti mō x (complex solution)

$$x=\frac{3+\sqrt{131}i}{10}\approx 0.3+1.144552314i$$

$$x=\frac{-\sqrt{131}i+3}{10}\approx 0.3-1.144552314i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 3x mai i ngā taha e rua.

$$5x^{2}-3x=-7$$

Me tāpiri te 7 ki ngā taha e rua.

$$5x^{2}-3x+7=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 5 mō a, -3 mō b, me 7 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 5\times 7}}{2\times 5}$$

Pūrua -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\times 5\times 7}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$x=\frac{-\left(-3\right)±\sqrt{9-20\times 7}}{2\times 5}$$

Whakareatia -20 ki te 7.

$$x=\frac{-\left(-3\right)±\sqrt{9-140}}{2\times 5}$$

Tāpiri 9 ki te -140.

$$x=\frac{-\left(-3\right)±\sqrt{-131}}{2\times 5}$$

Tuhia te pūtakerua o te -131.

$$x=\frac{-\left(-3\right)±\sqrt{131}i}{2\times 5}$$

Ko te tauaro o -3 ko 3.

$$x=\frac{3±\sqrt{131}i}{2\times 5}$$

Whakareatia 2 ki te 5.

$$x=\frac{3±\sqrt{131}i}{10}$$

Nā, me whakaoti te whārite x=\frac{3±\sqrt{131}i}{10} ina he tāpiri te ±. Tāpiri 3 ki te i\sqrt{131}.

$$x=\frac{3+\sqrt{131}i}{10}$$

Nā, me whakaoti te whārite x=\frac{3±\sqrt{131}i}{10} ina he tango te ±. Tango i\sqrt{131} mai i 3.

$$x=\frac{-\sqrt{131}i+3}{10}$$

Kua oti te whārite te whakatau.

$$x=\frac{3+\sqrt{131}i}{10} x=\frac{-\sqrt{131}i+3}{10}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te 3x mai i ngā taha e rua.

$$5x^{2}-3x=-7$$

Whakawehea ngā taha e rua ki te 5.

$$\frac{5x^{2}-3x}{5}=-\frac{7}{5}$$

Mā te whakawehe ki te 5 ka wetekia te whakareanga ki te 5.

$$x^{2}-\frac{3}{5}x=-\frac{7}{5}$$

Whakawehea te -\frac{3}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{3}{10}. Nā, tāpiria te pūrua o te -\frac{3}{10} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{3}{5}x+\left(-\frac{3}{10}\right)^{2}=-\frac{7}{5}+\left(-\frac{3}{10}\right)^{2}$$

Pūruatia -\frac{3}{10} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{3}{5}x+\frac{9}{100}=-\frac{7}{5}+\frac{9}{100}$$

Tāpiri -\frac{7}{5} ki te \frac{9}{100} 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.

$$x^{2}-\frac{3}{5}x+\frac{9}{100}=-\frac{131}{100}$$

Tauwehea x^{2}-\frac{3}{5}x+\frac{9}{100}. 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}.

$$\left(x-\frac{3}{10}\right)^{2}=-\frac{131}{100}$$

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

$$\sqrt{\left(x-\frac{3}{10}\right)^{2}}=\sqrt{-\frac{131}{100}}$$

Whakarūnātia.

$$x-\frac{3}{10}=\frac{\sqrt{131}i}{10} x-\frac{3}{10}=-\frac{\sqrt{131}i}{10}$$

Me tāpiri \frac{3}{10} ki ngā taha e rua o te whārite.

$$x=\frac{3+\sqrt{131}i}{10} x=\frac{-\sqrt{131}i+3}{10}$$