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

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

Whakaoti mō x (complex solution)

$$x=\frac{-\sqrt{51}i+3}{10}\approx 0.3-0.714142843i$$

$$x=\frac{3+\sqrt{51}i}{10}\approx 0.3+0.714142843i$$

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.

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

Me tango 3 mai i ngā taha e rua o te whārite.

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

Mā te tango i te 3 i a ia ake anō ka toe ko te 0.

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

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

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

Pūrua 3.

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

Whakareatia -4 ki te -5.

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

Whakareatia 20 ki te -3.

$$x=\frac{-3±\sqrt{9-60}}{2\left(-5\right)}$$

Tāpiri 9 ki te -60.

$$x=\frac{-3±\sqrt{-51}}{2\left(-5\right)}$$

Tuhia te pūtakerua o te -51.

$$x=\frac{-3±\sqrt{51}i}{2\left(-5\right)}$$

Whakareatia 2 ki te -5.

$$x=\frac{-3±\sqrt{51}i}{-10}$$

Nā, me whakaoti te whārite x=\frac{-3±\sqrt{51}i}{-10} ina he tāpiri te ±. Tāpiri -3 ki te i\sqrt{51}.

$$x=\frac{-3+\sqrt{51}i}{-10}$$

Whakawehe -3+i\sqrt{51} ki te -10.

$$x=\frac{-\sqrt{51}i+3}{10}$$

Nā, me whakaoti te whārite x=\frac{-3±\sqrt{51}i}{-10} ina he tango te ±. Tango i\sqrt{51} mai i -3.

$$x=\frac{-\sqrt{51}i-3}{-10}$$

Whakawehe -3-i\sqrt{51} ki te -10.

$$x=\frac{3+\sqrt{51}i}{10}$$

Kua oti te whārite te whakatau.

$$x=\frac{-\sqrt{51}i+3}{10} x=\frac{3+\sqrt{51}i}{10}$$

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.

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

Whakawehea ngā taha e rua ki te -5.

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

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

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

Whakawehe 3 ki te -5.

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

Whakawehe 3 ki te -5.

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

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.

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

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

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

Tāpiri -\frac{3}{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.

$$x^{2}-\frac{3}{5}x+\frac{9}{100}=-\frac{51}{100}$$

Tauwehea te x^{2}-\frac{3}{5}x+\frac{9}{100}. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{3}{10}\right)^{2}=-\frac{51}{100}$$

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

$$\sqrt{\left(x-\frac{3}{10}\right)^{2}}=\sqrt{-\frac{51}{100}}$$

Whakarūnātia.

$$x-\frac{3}{10}=\frac{\sqrt{51}i}{10} x-\frac{3}{10}=-\frac{\sqrt{51}i}{10}$$

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

$$x=\frac{3+\sqrt{51}i}{10} x=\frac{-\sqrt{51}i+3}{10}$$