Whakaoti i te 6x^2-13x+39=0

$$6 x ^ { 2 } - 13 x + 39 = 0$$

Whakaoti mō x (complex solution)

$$x=\frac{13+\sqrt{767}i}{12}\approx 1.083333333+2.307897071i$$

$$x=\frac{-\sqrt{767}i+13}{12}\approx 1.083333333-2.307897071i$$

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.

$$6x^{2}-13x+39=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 6 mō a, -13 mō b, me 39 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\times 6\times 39}}{2\times 6}$$

Pūrua -13.

$$x=\frac{-\left(-13\right)±\sqrt{169-4\times 6\times 39}}{2\times 6}$$

Whakareatia -4 ki te 6.

$$x=\frac{-\left(-13\right)±\sqrt{169-24\times 39}}{2\times 6}$$

Whakareatia -24 ki te 39.

$$x=\frac{-\left(-13\right)±\sqrt{169-936}}{2\times 6}$$

Tāpiri 169 ki te -936.

$$x=\frac{-\left(-13\right)±\sqrt{-767}}{2\times 6}$$

Tuhia te pūtakerua o te -767.

$$x=\frac{-\left(-13\right)±\sqrt{767}i}{2\times 6}$$

Ko te tauaro o -13 ko 13.

$$x=\frac{13±\sqrt{767}i}{2\times 6}$$

Whakareatia 2 ki te 6.

$$x=\frac{13±\sqrt{767}i}{12}$$

Nā, me whakaoti te whārite x=\frac{13±\sqrt{767}i}{12} ina he tāpiri te ±. Tāpiri 13 ki te i\sqrt{767}.

$$x=\frac{13+\sqrt{767}i}{12}$$

Nā, me whakaoti te whārite x=\frac{13±\sqrt{767}i}{12} ina he tango te ±. Tango i\sqrt{767} mai i 13.

$$x=\frac{-\sqrt{767}i+13}{12}$$

Kua oti te whārite te whakatau.

$$x=\frac{13+\sqrt{767}i}{12} x=\frac{-\sqrt{767}i+13}{12}$$

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.

$$6x^{2}-13x+39=0$$

Me tango 39 mai i ngā taha e rua o te whārite.

$$6x^{2}-13x+39-39=-39$$

Mā te tango i te 39 i a ia ake anō ka toe ko te 0.

$$6x^{2}-13x=-39$$

Whakawehea ngā taha e rua ki te 6.

$$\frac{6x^{2}-13x}{6}=-\frac{39}{6}$$

Mā te whakawehe ki te 6 ka wetekia te whakareanga ki te 6.

$$x^{2}-\frac{13}{6}x=-\frac{39}{6}$$

Whakahekea te hautanga \frac{-39}{6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 3.

$$x^{2}-\frac{13}{6}x=-\frac{13}{2}$$

Whakawehea te -\frac{13}{6}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{13}{12}. Nā, tāpiria te pūrua o te -\frac{13}{12} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{13}{6}x+\left(-\frac{13}{12}\right)^{2}=-\frac{13}{2}+\left(-\frac{13}{12}\right)^{2}$$

Pūruatia -\frac{13}{12} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{13}{6}x+\frac{169}{144}=-\frac{13}{2}+\frac{169}{144}$$

Tāpiri -\frac{13}{2} ki te \frac{169}{144} 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.

$$x^{2}-\frac{13}{6}x+\frac{169}{144}=-\frac{767}{144}$$

Tauwehea x^{2}-\frac{13}{6}x+\frac{169}{144}. 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}.

$$\left(x-\frac{13}{12}\right)^{2}=-\frac{767}{144}$$

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

$$\sqrt{\left(x-\frac{13}{12}\right)^{2}}=\sqrt{-\frac{767}{144}}$$

Whakarūnātia.

$$x-\frac{13}{12}=\frac{\sqrt{767}i}{12} x-\frac{13}{12}=-\frac{\sqrt{767}i}{12}$$

Me tāpiri \frac{13}{12} ki ngā taha e rua o te whārite.

$$x=\frac{13+\sqrt{767}i}{12} x=\frac{-\sqrt{767}i+13}{12}$$