Whakaoti i te 2x^2+3/8x+16=0

$$2 x ^ { 2 } + \frac { 3 } { 8 } x + 16 = 0$$

Whakaoti mō x (complex solution)

$$x=\frac{-3+7\sqrt{167}i}{32}\approx -0.09375+2.826872996i$$

$$x=\frac{-7\sqrt{167}i-3}{32}\approx -0.09375-2.826872996i$$

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.

$$2x^{2}+\frac{3}{8}x+16=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, \frac{3}{8} mō b, me 16 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{3}{8}±\sqrt{\left(\frac{3}{8}\right)^{2}-4\times 2\times 16}}{2\times 2}$$

Pūruatia \frac{3}{8} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x=\frac{-\frac{3}{8}±\sqrt{\frac{9}{64}-4\times 2\times 16}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-\frac{3}{8}±\sqrt{\frac{9}{64}-8\times 16}}{2\times 2}$$

Whakareatia -8 ki te 16.

$$x=\frac{-\frac{3}{8}±\sqrt{\frac{9}{64}-128}}{2\times 2}$$

Tāpiri \frac{9}{64} ki te -128.

$$x=\frac{-\frac{3}{8}±\sqrt{-\frac{8183}{64}}}{2\times 2}$$

Tuhia te pūtakerua o te -\frac{8183}{64}.

$$x=\frac{-\frac{3}{8}±\frac{7\sqrt{167}i}{8}}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{-\frac{3}{8}±\frac{7\sqrt{167}i}{8}}{4}$$

Nā, me whakaoti te whārite x=\frac{-\frac{3}{8}±\frac{7\sqrt{167}i}{8}}{4} ina he tāpiri te ±. Tāpiri -\frac{3}{8} ki te \frac{7i\sqrt{167}}{8}.

$$x=\frac{-3+7\sqrt{167}i}{4\times 8}$$

Whakawehe \frac{-3+7i\sqrt{167}}{8} ki te 4.

$$x=\frac{-3+7\sqrt{167}i}{32}$$

Nā, me whakaoti te whārite x=\frac{-\frac{3}{8}±\frac{7\sqrt{167}i}{8}}{4} ina he tango te ±. Tango \frac{7i\sqrt{167}}{8} mai i -\frac{3}{8}.

$$x=\frac{-7\sqrt{167}i-3}{4\times 8}$$

Whakawehe \frac{-3-7i\sqrt{167}}{8} ki te 4.

$$x=\frac{-7\sqrt{167}i-3}{32}$$

Kua oti te whārite te whakatau.

$$x=\frac{-3+7\sqrt{167}i}{32} x=\frac{-7\sqrt{167}i-3}{32}$$

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.

$$2x^{2}+\frac{3}{8}x+16=0$$

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

$$2x^{2}+\frac{3}{8}x+16-16=-16$$

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

$$2x^{2}+\frac{3}{8}x=-16$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}+\frac{3}{8}x}{2}=-\frac{16}{2}$$

Mā te whakawehe ki te 2 ka wetekia te whakareanga ki te 2.

$$x^{2}+\frac{\frac{3}{8}}{2}x=-\frac{16}{2}$$

Whakawehe \frac{3}{8} ki te 2.

$$x^{2}+\frac{3}{16}x=-\frac{16}{2}$$

Whakawehe -16 ki te 2.

$$x^{2}+\frac{3}{16}x=-8$$

Whakawehea te \frac{3}{16}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{3}{32}. Nā, tāpiria te pūrua o te \frac{3}{32} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{3}{16}x+\left(\frac{3}{32}\right)^{2}=-8+\left(\frac{3}{32}\right)^{2}$$

Pūruatia \frac{3}{32} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{3}{16}x+\frac{9}{1024}=-8+\frac{9}{1024}$$

Tāpiri -8 ki te \frac{9}{1024}.

$$x^{2}+\frac{3}{16}x+\frac{9}{1024}=-\frac{8183}{1024}$$

Tauwehea x^{2}+\frac{3}{16}x+\frac{9}{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}.

$$\left(x+\frac{3}{32}\right)^{2}=-\frac{8183}{1024}$$

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

$$\sqrt{\left(x+\frac{3}{32}\right)^{2}}=\sqrt{-\frac{8183}{1024}}$$

Whakarūnātia.

$$x+\frac{3}{32}=\frac{7\sqrt{167}i}{32} x+\frac{3}{32}=-\frac{7\sqrt{167}i}{32}$$

Me tango \frac{3}{32} mai i ngā taha e rua o te whārite.

$$x=\frac{-3+7\sqrt{167}i}{32} x=\frac{-7\sqrt{167}i-3}{32}$$