Whakaoti i te -144x^2+9x-9=0

$$-144 { x }^{ 2 } +9x-9 = 0$$

Whakaoti mō x (complex solution)

$$x=\frac{1+3\sqrt{7}i}{32}\approx 0.03125+0.248039185i$$

$$x=\frac{-3\sqrt{7}i+1}{32}\approx 0.03125-0.248039185i$$

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.

$$-144x^{2}+9x-9=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -144 mō a, 9 mō b, me -9 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-9±\sqrt{9^{2}-4\left(-144\right)\left(-9\right)}}{2\left(-144\right)}$$

Pūrua 9.

$$x=\frac{-9±\sqrt{81-4\left(-144\right)\left(-9\right)}}{2\left(-144\right)}$$

Whakareatia -4 ki te -144.

$$x=\frac{-9±\sqrt{81+576\left(-9\right)}}{2\left(-144\right)}$$

Whakareatia 576 ki te -9.

$$x=\frac{-9±\sqrt{81-5184}}{2\left(-144\right)}$$

Tāpiri 81 ki te -5184.

$$x=\frac{-9±\sqrt{-5103}}{2\left(-144\right)}$$

Tuhia te pūtakerua o te -5103.

$$x=\frac{-9±27\sqrt{7}i}{2\left(-144\right)}$$

Whakareatia 2 ki te -144.

$$x=\frac{-9±27\sqrt{7}i}{-288}$$

Nā, me whakaoti te whārite x=\frac{-9±27\sqrt{7}i}{-288} ina he tāpiri te ±. Tāpiri -9 ki te 27i\sqrt{7}.

$$x=\frac{-9+27\sqrt{7}i}{-288}$$

Whakawehe -9+27i\sqrt{7} ki te -288.

$$x=\frac{-3\sqrt{7}i+1}{32}$$

Nā, me whakaoti te whārite x=\frac{-9±27\sqrt{7}i}{-288} ina he tango te ±. Tango 27i\sqrt{7} mai i -9.

$$x=\frac{-27\sqrt{7}i-9}{-288}$$

Whakawehe -9-27i\sqrt{7} ki te -288.

$$x=\frac{1+3\sqrt{7}i}{32}$$

Kua oti te whārite te whakatau.

$$x=\frac{-3\sqrt{7}i+1}{32} x=\frac{1+3\sqrt{7}i}{32}$$

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.

$$-144x^{2}+9x-9=0$$

Me tāpiri 9 ki ngā taha e rua o te whārite.

$$-144x^{2}+9x-9-\left(-9\right)=-\left(-9\right)$$

Mā te tango i te -9 i a ia ake anō ka toe ko te 0.

$$-144x^{2}+9x=-\left(-9\right)$$

Tango -9 mai i 0.

$$-144x^{2}+9x=9$$

Whakawehea ngā taha e rua ki te -144.

$$\frac{-144x^{2}+9x}{-144}=\frac{9}{-144}$$

Mā te whakawehe ki te -144 ka wetekia te whakareanga ki te -144.

$$x^{2}+\frac{9}{-144}x=\frac{9}{-144}$$

Whakahekea te hautanga \frac{9}{-144} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 9.

$$x^{2}-\frac{1}{16}x=\frac{9}{-144}$$

Whakahekea te hautanga \frac{9}{-144} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 9.

$$x^{2}-\frac{1}{16}x=-\frac{1}{16}$$

Whakawehea te -\frac{1}{16}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{32}. Nā, tāpiria te pūrua o te -\frac{1}{32} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{1}{16}x+\left(-\frac{1}{32}\right)^{2}=-\frac{1}{16}+\left(-\frac{1}{32}\right)^{2}$$

Pūruatia -\frac{1}{32} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{1}{16}x+\frac{1}{1024}=-\frac{1}{16}+\frac{1}{1024}$$

Tāpiri -\frac{1}{16} ki te \frac{1}{1024} 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.

$$x^{2}-\frac{1}{16}x+\frac{1}{1024}=-\frac{63}{1024}$$

Tauwehea x^{2}-\frac{1}{16}x+\frac{1}{1024}. 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}.

$$\left(x-\frac{1}{32}\right)^{2}=-\frac{63}{1024}$$

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

$$\sqrt{\left(x-\frac{1}{32}\right)^{2}}=\sqrt{-\frac{63}{1024}}$$

Whakarūnātia.

$$x-\frac{1}{32}=\frac{3\sqrt{7}i}{32} x-\frac{1}{32}=-\frac{3\sqrt{7}i}{32}$$

Me tāpiri \frac{1}{32} ki ngā taha e rua o te whārite.

$$x=\frac{1+3\sqrt{7}i}{32} x=\frac{-3\sqrt{7}i+1}{32}$$