Whakaoti i te 5x^2-2x+15=0

$$5 { x }^{ 2 } -2x+15=0$$

Whakaoti mō x (complex solution)

$$x=\frac{1+\sqrt{74}i}{5}\approx 0.2+1.720465053i$$

$$x=\frac{-\sqrt{74}i+1}{5}\approx 0.2-1.720465053i$$

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.

$$5x^{2}-2x+15=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 5 mō a, -2 mō b, me 15 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 5\times 15}}{2\times 5}$$

Pūrua -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\times 5\times 15}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$x=\frac{-\left(-2\right)±\sqrt{4-20\times 15}}{2\times 5}$$

Whakareatia -20 ki te 15.

$$x=\frac{-\left(-2\right)±\sqrt{4-300}}{2\times 5}$$

Tāpiri 4 ki te -300.

$$x=\frac{-\left(-2\right)±\sqrt{-296}}{2\times 5}$$

Tuhia te pūtakerua o te -296.

$$x=\frac{-\left(-2\right)±2\sqrt{74}i}{2\times 5}$$

Ko te tauaro o -2 ko 2.

$$x=\frac{2±2\sqrt{74}i}{2\times 5}$$

Whakareatia 2 ki te 5.

$$x=\frac{2±2\sqrt{74}i}{10}$$

Nā, me whakaoti te whārite x=\frac{2±2\sqrt{74}i}{10} ina he tāpiri te ±. Tāpiri 2 ki te 2i\sqrt{74}.

$$x=\frac{2+2\sqrt{74}i}{10}$$

Whakawehe 2+2i\sqrt{74} ki te 10.

$$x=\frac{1+\sqrt{74}i}{5}$$

Nā, me whakaoti te whārite x=\frac{2±2\sqrt{74}i}{10} ina he tango te ±. Tango 2i\sqrt{74} mai i 2.

$$x=\frac{-2\sqrt{74}i+2}{10}$$

Whakawehe 2-2i\sqrt{74} ki te 10.

$$x=\frac{-\sqrt{74}i+1}{5}$$

Kua oti te whārite te whakatau.

$$x=\frac{1+\sqrt{74}i}{5} x=\frac{-\sqrt{74}i+1}{5}$$

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.

$$5x^{2}-2x+15=0$$

Me tango 15 mai i ngā taha e rua o te whārite.

$$5x^{2}-2x+15-15=-15$$

Mā te tango i te 15 i a ia ake anō ka toe ko te 0.

$$5x^{2}-2x=-15$$

Whakawehea ngā taha e rua ki te 5.

$$\frac{5x^{2}-2x}{5}=-\frac{15}{5}$$

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

$$x^{2}-\frac{2}{5}x=-\frac{15}{5}$$

Whakawehe -15 ki te 5.

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

Whakawehea te -\frac{2}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{5}. Nā, tāpiria te pūrua o te -\frac{1}{5} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{2}{5}x+\left(-\frac{1}{5}\right)^{2}=-3+\left(-\frac{1}{5}\right)^{2}$$

Pūruatia -\frac{1}{5} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{2}{5}x+\frac{1}{25}=-3+\frac{1}{25}$$

Tāpiri -3 ki te \frac{1}{25}.

$$x^{2}-\frac{2}{5}x+\frac{1}{25}=-\frac{74}{25}$$

Tauwehea x^{2}-\frac{2}{5}x+\frac{1}{25}. 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}.

$$\left(x-\frac{1}{5}\right)^{2}=-\frac{74}{25}$$

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

$$\sqrt{\left(x-\frac{1}{5}\right)^{2}}=\sqrt{-\frac{74}{25}}$$

Whakarūnātia.

$$x-\frac{1}{5}=\frac{\sqrt{74}i}{5} x-\frac{1}{5}=-\frac{\sqrt{74}i}{5}$$

Me tāpiri \frac{1}{5} ki ngā taha e rua o te whārite.

$$x=\frac{1+\sqrt{74}i}{5} x=\frac{-\sqrt{74}i+1}{5}$$