Whakaoti i te left(2xright)^2+5x+6=0

$${ \left(2x \right) }^{ 2 } +5x+6=0$$

Whakaoti mō x (complex solution)

$$x=\frac{-5+\sqrt{71}i}{8}\approx -0.625+1.053268722i$$

$$x=\frac{-\sqrt{71}i-5}{8}\approx -0.625-1.053268722i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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

$$2^{2}x^{2}+5x+6=0$$

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

$$4x^{2}+5x+6=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, 5 mō b, me 6 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\times 4\times 6}}{2\times 4}$$

Pūrua 5.

$$x=\frac{-5±\sqrt{25-4\times 4\times 6}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$x=\frac{-5±\sqrt{25-16\times 6}}{2\times 4}$$

Whakareatia -16 ki te 6.

$$x=\frac{-5±\sqrt{25-96}}{2\times 4}$$

Tāpiri 25 ki te -96.

$$x=\frac{-5±\sqrt{-71}}{2\times 4}$$

Tuhia te pūtakerua o te -71.

$$x=\frac{-5±\sqrt{71}i}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{-5±\sqrt{71}i}{8}$$

Nā, me whakaoti te whārite x=\frac{-5±\sqrt{71}i}{8} ina he tāpiri te ±. Tāpiri -5 ki te i\sqrt{71}.

$$x=\frac{-5+\sqrt{71}i}{8}$$

Nā, me whakaoti te whārite x=\frac{-5±\sqrt{71}i}{8} ina he tango te ±. Tango i\sqrt{71} mai i -5.

$$x=\frac{-\sqrt{71}i-5}{8}$$

Kua oti te whārite te whakatau.

$$x=\frac{-5+\sqrt{71}i}{8} x=\frac{-\sqrt{71}i-5}{8}$$

Ngā Upane mō te Whakaoti i te Pūrua

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

$$2^{2}x^{2}+5x+6=0$$

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

$$4x^{2}+5x+6=0$$

Tangohia te 6 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$4x^{2}+5x=-6$$

Whakawehea ngā taha e rua ki te 4.

$$\frac{4x^{2}+5x}{4}=-\frac{6}{4}$$

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

$$x^{2}+\frac{5}{4}x=-\frac{6}{4}$$

Whakahekea te hautanga \frac{-6}{4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}+\frac{5}{4}x=-\frac{3}{2}$$

Whakawehea te \frac{5}{4}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{5}{8}. Nā, tāpiria te pūrua o te \frac{5}{8} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{5}{4}x+\left(\frac{5}{8}\right)^{2}=-\frac{3}{2}+\left(\frac{5}{8}\right)^{2}$$

Pūruatia \frac{5}{8} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{5}{4}x+\frac{25}{64}=-\frac{3}{2}+\frac{25}{64}$$

Tāpiri -\frac{3}{2} ki te \frac{25}{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.

$$x^{2}+\frac{5}{4}x+\frac{25}{64}=-\frac{71}{64}$$

Tauwehea x^{2}+\frac{5}{4}x+\frac{25}{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}.

$$\left(x+\frac{5}{8}\right)^{2}=-\frac{71}{64}$$

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

$$\sqrt{\left(x+\frac{5}{8}\right)^{2}}=\sqrt{-\frac{71}{64}}$$

Whakarūnātia.

$$x+\frac{5}{8}=\frac{\sqrt{71}i}{8} x+\frac{5}{8}=-\frac{\sqrt{71}i}{8}$$

Me tango \frac{5}{8} mai i ngā taha e rua o te whārite.

$$x=\frac{-5+\sqrt{71}i}{8} x=\frac{-\sqrt{71}i-5}{8}$$