Whakaoti i te 4x^2-11x+30=16

$$4 x ^ { 2 } - 11 x + 30 = 16$$

Whakaoti mō x (complex solution)

$$x=\frac{11+\sqrt{103}i}{8}\approx 1.375+1.268611446i$$

$$x=\frac{-\sqrt{103}i+11}{8}\approx 1.375-1.268611446i$$

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.

$$4x^{2}-11x+30=16$$

Me tango 16 mai i ngā taha e rua o te whārite.

$$4x^{2}-11x+30-16=16-16$$

Mā te tango i te 16 i a ia ake anō ka toe ko te 0.

$$4x^{2}-11x+30-16=0$$

Tango 16 mai i 30.

$$4x^{2}-11x+14=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, -11 mō b, me 14 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\times 4\times 14}}{2\times 4}$$

Pūrua -11.

$$x=\frac{-\left(-11\right)±\sqrt{121-4\times 4\times 14}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$x=\frac{-\left(-11\right)±\sqrt{121-16\times 14}}{2\times 4}$$

Whakareatia -16 ki te 14.

$$x=\frac{-\left(-11\right)±\sqrt{121-224}}{2\times 4}$$

Tāpiri 121 ki te -224.

$$x=\frac{-\left(-11\right)±\sqrt{-103}}{2\times 4}$$

Tuhia te pūtakerua o te -103.

$$x=\frac{-\left(-11\right)±\sqrt{103}i}{2\times 4}$$

Ko te tauaro o -11 ko 11.

$$x=\frac{11±\sqrt{103}i}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{11±\sqrt{103}i}{8}$$

Nā, me whakaoti te whārite x=\frac{11±\sqrt{103}i}{8} ina he tāpiri te ±. Tāpiri 11 ki te i\sqrt{103}.

$$x=\frac{11+\sqrt{103}i}{8}$$

Nā, me whakaoti te whārite x=\frac{11±\sqrt{103}i}{8} ina he tango te ±. Tango i\sqrt{103} mai i 11.

$$x=\frac{-\sqrt{103}i+11}{8}$$

Kua oti te whārite te whakatau.

$$x=\frac{11+\sqrt{103}i}{8} x=\frac{-\sqrt{103}i+11}{8}$$

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.

$$4x^{2}-11x+30=16$$

Me tango 30 mai i ngā taha e rua o te whārite.

$$4x^{2}-11x+30-30=16-30$$

Mā te tango i te 30 i a ia ake anō ka toe ko te 0.

$$4x^{2}-11x=16-30$$

Tango 30 mai i 16.

$$4x^{2}-11x=-14$$

Whakawehea ngā taha e rua ki te 4.

$$\frac{4x^{2}-11x}{4}=-\frac{14}{4}$$

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

$$x^{2}-\frac{11}{4}x=-\frac{14}{4}$$

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

$$x^{2}-\frac{11}{4}x=-\frac{7}{2}$$

Whakawehea te -\frac{11}{4}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{11}{8}. Nā, tāpiria te pūrua o te -\frac{11}{8} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{11}{4}x+\left(-\frac{11}{8}\right)^{2}=-\frac{7}{2}+\left(-\frac{11}{8}\right)^{2}$$

Pūruatia -\frac{11}{8} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{11}{4}x+\frac{121}{64}=-\frac{7}{2}+\frac{121}{64}$$

Tāpiri -\frac{7}{2} ki te \frac{121}{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.

$$x^{2}-\frac{11}{4}x+\frac{121}{64}=-\frac{103}{64}$$

Tauwehea te x^{2}-\frac{11}{4}x+\frac{121}{64}. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{11}{8}\right)^{2}=-\frac{103}{64}$$

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

$$\sqrt{\left(x-\frac{11}{8}\right)^{2}}=\sqrt{-\frac{103}{64}}$$

Whakarūnātia.

$$x-\frac{11}{8}=\frac{\sqrt{103}i}{8} x-\frac{11}{8}=-\frac{\sqrt{103}i}{8}$$

Me tāpiri \frac{11}{8} ki ngā taha e rua o te whārite.

$$x=\frac{11+\sqrt{103}i}{8} x=\frac{-\sqrt{103}i+11}{8}$$