Whakaoti i te 4x^2-5x+10=0

$$4 { x }^{ 2 } -5x+10=0$$

Whakaoti mō x (complex solution)

$$x=\frac{5+3\sqrt{15}i}{8}\approx 0.625+1.452368755i$$

$$x=\frac{-3\sqrt{15}i+5}{8}\approx 0.625-1.452368755i$$

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.

$$4x^{2}-5x+10=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, -5 mō b, me 10 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 4\times 10}}{2\times 4}$$

Pūrua -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\times 4\times 10}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$x=\frac{-\left(-5\right)±\sqrt{25-16\times 10}}{2\times 4}$$

Whakareatia -16 ki te 10.

$$x=\frac{-\left(-5\right)±\sqrt{25-160}}{2\times 4}$$

Tāpiri 25 ki te -160.

$$x=\frac{-\left(-5\right)±\sqrt{-135}}{2\times 4}$$

Tuhia te pūtakerua o te -135.

$$x=\frac{-\left(-5\right)±3\sqrt{15}i}{2\times 4}$$

Ko te tauaro o -5 ko 5.

$$x=\frac{5±3\sqrt{15}i}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{5±3\sqrt{15}i}{8}$$

Nā, me whakaoti te whārite x=\frac{5±3\sqrt{15}i}{8} ina he tāpiri te ±. Tāpiri 5 ki te 3i\sqrt{15}.

$$x=\frac{5+3\sqrt{15}i}{8}$$

Nā, me whakaoti te whārite x=\frac{5±3\sqrt{15}i}{8} ina he tango te ±. Tango 3i\sqrt{15} mai i 5.

$$x=\frac{-3\sqrt{15}i+5}{8}$$

Kua oti te whārite te whakatau.

$$x=\frac{5+3\sqrt{15}i}{8} x=\frac{-3\sqrt{15}i+5}{8}$$

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.

$$4x^{2}-5x+10=0$$

Me tango 10 mai i ngā taha e rua o te whārite.

$$4x^{2}-5x+10-10=-10$$

Mā te tango i te 10 i a ia ake anō ka toe ko te 0.

$$4x^{2}-5x=-10$$

Whakawehea ngā taha e rua ki te 4.

$$\frac{4x^{2}-5x}{4}=-\frac{10}{4}$$

Mā te whakawehe ki te 4 ka wetekia te whakareanga ki te 4.

$$x^{2}-\frac{5}{4}x=-\frac{10}{4}$$

Whakahekea te hautanga \frac{-10}{4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}-\frac{5}{4}x=-\frac{5}{2}$$

Whakawehea te -\frac{5}{4}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{5}{8}. Nā, tāpiria te pūrua o te -\frac{5}{8} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{5}{4}x+\left(-\frac{5}{8}\right)^{2}=-\frac{5}{2}+\left(-\frac{5}{8}\right)^{2}$$

Pūruatia -\frac{5}{8} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{5}{4}x+\frac{25}{64}=-\frac{5}{2}+\frac{25}{64}$$

Tāpiri -\frac{5}{2} ki te \frac{25}{64} 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.

$$x^{2}-\frac{5}{4}x+\frac{25}{64}=-\frac{135}{64}$$

Tauwehea x^{2}-\frac{5}{4}x+\frac{25}{64}. 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}.

$$\left(x-\frac{5}{8}\right)^{2}=-\frac{135}{64}$$

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

$$\sqrt{\left(x-\frac{5}{8}\right)^{2}}=\sqrt{-\frac{135}{64}}$$

Whakarūnātia.

$$x-\frac{5}{8}=\frac{3\sqrt{15}i}{8} x-\frac{5}{8}=-\frac{3\sqrt{15}i}{8}$$

Me tāpiri \frac{5}{8} ki ngā taha e rua o te whārite.

$$x=\frac{5+3\sqrt{15}i}{8} x=\frac{-3\sqrt{15}i+5}{8}$$