Whakaoti i te 3left(2xright)^2+5x+30=0

$$3 { \left(2x \right) }^{ 2 } +5x+30 = 0$$

Whakaoti mō x (complex solution)

$$x=\frac{-5+\sqrt{1415}i}{24}\approx -0.208333333+1.567353573i$$

$$x=\frac{-\sqrt{1415}i-5}{24}\approx -0.208333333-1.567353573i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakarohaina te \left(2x\right)^{2}.

$$3\times 2^{2}x^{2}+5x+30=0$$

Tātaihia te 2 mā te pū o 2, kia riro ko 4.

$$3\times 4x^{2}+5x+30=0$$

Whakareatia te 3 ki te 4, ka 12.

$$12x^{2}+5x+30=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 12 mō a, 5 mō b, me 30 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\times 12\times 30}}{2\times 12}$$

Pūrua 5.

$$x=\frac{-5±\sqrt{25-4\times 12\times 30}}{2\times 12}$$

Whakareatia -4 ki te 12.

$$x=\frac{-5±\sqrt{25-48\times 30}}{2\times 12}$$

Whakareatia -48 ki te 30.

$$x=\frac{-5±\sqrt{25-1440}}{2\times 12}$$

Tāpiri 25 ki te -1440.

$$x=\frac{-5±\sqrt{-1415}}{2\times 12}$$

Tuhia te pūtakerua o te -1415.

$$x=\frac{-5±\sqrt{1415}i}{2\times 12}$$

Whakareatia 2 ki te 12.

$$x=\frac{-5±\sqrt{1415}i}{24}$$

Nā, me whakaoti te whārite x=\frac{-5±\sqrt{1415}i}{24} ina he tāpiri te ±. Tāpiri -5 ki te i\sqrt{1415}.

$$x=\frac{-5+\sqrt{1415}i}{24}$$

Nā, me whakaoti te whārite x=\frac{-5±\sqrt{1415}i}{24} ina he tango te ±. Tango i\sqrt{1415} mai i -5.

$$x=\frac{-\sqrt{1415}i-5}{24}$$

Kua oti te whārite te whakatau.

$$x=\frac{-5+\sqrt{1415}i}{24} x=\frac{-\sqrt{1415}i-5}{24}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakarohaina te \left(2x\right)^{2}.

$$3\times 2^{2}x^{2}+5x+30=0$$

Tātaihia te 2 mā te pū o 2, kia riro ko 4.

$$3\times 4x^{2}+5x+30=0$$

Whakareatia te 3 ki te 4, ka 12.

$$12x^{2}+5x+30=0$$

Tangohia te 30 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$12x^{2}+5x=-30$$

Whakawehea ngā taha e rua ki te 12.

$$\frac{12x^{2}+5x}{12}=-\frac{30}{12}$$

Mā te whakawehe ki te 12 ka wetekia te whakareanga ki te 12.

$$x^{2}+\frac{5}{12}x=-\frac{30}{12}$$

Whakahekea te hautanga \frac{-30}{12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$x^{2}+\frac{5}{12}x=-\frac{5}{2}$$

Whakawehea te \frac{5}{12}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{5}{24}. Nā, tāpiria te pūrua o te \frac{5}{24} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{5}{12}x+\left(\frac{5}{24}\right)^{2}=-\frac{5}{2}+\left(\frac{5}{24}\right)^{2}$$

Pūruatia \frac{5}{24} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{5}{12}x+\frac{25}{576}=-\frac{5}{2}+\frac{25}{576}$$

Tāpiri -\frac{5}{2} ki te \frac{25}{576} 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.

$$x^{2}+\frac{5}{12}x+\frac{25}{576}=-\frac{1415}{576}$$

Tauwehea x^{2}+\frac{5}{12}x+\frac{25}{576}. 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}.

$$\left(x+\frac{5}{24}\right)^{2}=-\frac{1415}{576}$$

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

$$\sqrt{\left(x+\frac{5}{24}\right)^{2}}=\sqrt{-\frac{1415}{576}}$$

Whakarūnātia.

$$x+\frac{5}{24}=\frac{\sqrt{1415}i}{24} x+\frac{5}{24}=-\frac{\sqrt{1415}i}{24}$$

Me tango \frac{5}{24} mai i ngā taha e rua o te whārite.

$$x=\frac{-5+\sqrt{1415}i}{24} x=\frac{-\sqrt{1415}i-5}{24}$$