Whakaoti i te 0=4x^2-9x+14

$$0 = 4 x ^ { 2 } - 9 x + 14$$

Whakaoti mō x (complex solution)

$$x=\frac{9+\sqrt{143}i}{8}\approx 1.125+1.494782593i$$

$$x=\frac{-\sqrt{143}i+9}{8}\approx 1.125-1.494782593i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$4x^{2}-9x+14=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, -9 mō b, me 14 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 4\times 14}}{2\times 4}$$

Pūrua -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\times 4\times 14}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$x=\frac{-\left(-9\right)±\sqrt{81-16\times 14}}{2\times 4}$$

Whakareatia -16 ki te 14.

$$x=\frac{-\left(-9\right)±\sqrt{81-224}}{2\times 4}$$

Tāpiri 81 ki te -224.

$$x=\frac{-\left(-9\right)±\sqrt{-143}}{2\times 4}$$

Tuhia te pūtakerua o te -143.

$$x=\frac{-\left(-9\right)±\sqrt{143}i}{2\times 4}$$

Ko te tauaro o -9 ko 9.

$$x=\frac{9±\sqrt{143}i}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{9±\sqrt{143}i}{8}$$

Nā, me whakaoti te whārite x=\frac{9±\sqrt{143}i}{8} ina he tāpiri te ±. Tāpiri 9 ki te i\sqrt{143}.

$$x=\frac{9+\sqrt{143}i}{8}$$

Nā, me whakaoti te whārite x=\frac{9±\sqrt{143}i}{8} ina he tango te ±. Tango i\sqrt{143} mai i 9.

$$x=\frac{-\sqrt{143}i+9}{8}$$

Kua oti te whārite te whakatau.

$$x=\frac{9+\sqrt{143}i}{8} x=\frac{-\sqrt{143}i+9}{8}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$4x^{2}-9x+14=0$$

Tangohia te 14 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$4x^{2}-9x=-14$$

Whakawehea ngā taha e rua ki te 4.

$$\frac{4x^{2}-9x}{4}=-\frac{14}{4}$$

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

$$x^{2}-\frac{9}{4}x=-\frac{14}{4}$$

Whakahekea te hautanga \frac{-14}{4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}-\frac{9}{4}x=-\frac{7}{2}$$

Whakawehea te -\frac{9}{4}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{9}{8}. Nā, tāpiria te pūrua o te -\frac{9}{8} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{9}{4}x+\left(-\frac{9}{8}\right)^{2}=-\frac{7}{2}+\left(-\frac{9}{8}\right)^{2}$$

Pūruatia -\frac{9}{8} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{9}{4}x+\frac{81}{64}=-\frac{7}{2}+\frac{81}{64}$$

Tāpiri -\frac{7}{2} ki te \frac{81}{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.

$$x^{2}-\frac{9}{4}x+\frac{81}{64}=-\frac{143}{64}$$

Tauwehea x^{2}-\frac{9}{4}x+\frac{81}{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}.

$$\left(x-\frac{9}{8}\right)^{2}=-\frac{143}{64}$$

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

$$\sqrt{\left(x-\frac{9}{8}\right)^{2}}=\sqrt{-\frac{143}{64}}$$

Whakarūnātia.

$$x-\frac{9}{8}=\frac{\sqrt{143}i}{8} x-\frac{9}{8}=-\frac{\sqrt{143}i}{8}$$

Me tāpiri \frac{9}{8} ki ngā taha e rua o te whārite.

$$x=\frac{9+\sqrt{143}i}{8} x=\frac{-\sqrt{143}i+9}{8}$$