Whakaoti i te 64x^2+24sqrt{5}x+33=0

$$64 x ^ { 2 } + 24 \sqrt { 5 } x + 33 = 0$$

Whakaoti mō x (complex solution)

$$x=\frac{-3\sqrt{5}+\sqrt{87}i}{16}\approx -0.419262746+0.582961191i$$

$$x=\frac{-\sqrt{87}i-3\sqrt{5}}{16}\approx -0.419262746-0.582961191i$$

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.

$$64x^{2}+24\sqrt{5}x+33=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 64 mō a, 24\sqrt{5} mō b, me 33 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-24\sqrt{5}±\sqrt{\left(24\sqrt{5}\right)^{2}-4\times 64\times 33}}{2\times 64}$$

Pūrua 24\sqrt{5}.

$$x=\frac{-24\sqrt{5}±\sqrt{2880-4\times 64\times 33}}{2\times 64}$$

Whakareatia -4 ki te 64.

$$x=\frac{-24\sqrt{5}±\sqrt{2880-256\times 33}}{2\times 64}$$

Whakareatia -256 ki te 33.

$$x=\frac{-24\sqrt{5}±\sqrt{2880-8448}}{2\times 64}$$

Tāpiri 2880 ki te -8448.

$$x=\frac{-24\sqrt{5}±\sqrt{-5568}}{2\times 64}$$

Tuhia te pūtakerua o te -5568.

$$x=\frac{-24\sqrt{5}±8\sqrt{87}i}{2\times 64}$$

Whakareatia 2 ki te 64.

$$x=\frac{-24\sqrt{5}±8\sqrt{87}i}{128}$$

Nā, me whakaoti te whārite x=\frac{-24\sqrt{5}±8\sqrt{87}i}{128} ina he tāpiri te ±. Tāpiri -24\sqrt{5} ki te 8i\sqrt{87}.

$$x=\frac{-24\sqrt{5}+8\sqrt{87}i}{128}$$

Whakawehe -24\sqrt{5}+8i\sqrt{87} ki te 128.

$$x=\frac{-3\sqrt{5}+\sqrt{87}i}{16}$$

Nā, me whakaoti te whārite x=\frac{-24\sqrt{5}±8\sqrt{87}i}{128} ina he tango te ±. Tango 8i\sqrt{87} mai i -24\sqrt{5}.

$$x=\frac{-8\sqrt{87}i-24\sqrt{5}}{128}$$

Whakawehe -24\sqrt{5}-8i\sqrt{87} ki te 128.

$$x=\frac{-\sqrt{87}i-3\sqrt{5}}{16}$$

Kua oti te whārite te whakatau.

$$x=\frac{-3\sqrt{5}+\sqrt{87}i}{16} x=\frac{-\sqrt{87}i-3\sqrt{5}}{16}$$

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.

$$64x^{2}+24\sqrt{5}x+33=0$$

Me tango 33 mai i ngā taha e rua o te whārite.

$$64x^{2}+24\sqrt{5}x+33-33=-33$$

Mā te tango i te 33 i a ia ake anō ka toe ko te 0.

$$64x^{2}+24\sqrt{5}x=-33$$

Whakawehea ngā taha e rua ki te 64.

$$\frac{64x^{2}+24\sqrt{5}x}{64}=-\frac{33}{64}$$

Mā te whakawehe ki te 64 ka wetekia te whakareanga ki te 64.

$$x^{2}+\frac{24\sqrt{5}}{64}x=-\frac{33}{64}$$

Whakawehe 24\sqrt{5} ki te 64.

$$x^{2}+\frac{3\sqrt{5}}{8}x=-\frac{33}{64}$$

Whakawehea te \frac{3\sqrt{5}}{8}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{3\sqrt{5}}{16}. Nā, tāpiria te pūrua o te \frac{3\sqrt{5}}{16} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{3\sqrt{5}}{8}x+\left(\frac{3\sqrt{5}}{16}\right)^{2}=-\frac{33}{64}+\left(\frac{3\sqrt{5}}{16}\right)^{2}$$

Pūrua \frac{3\sqrt{5}}{16}.

$$x^{2}+\frac{3\sqrt{5}}{8}x+\frac{45}{256}=-\frac{33}{64}+\frac{45}{256}$$

Tāpiri -\frac{33}{64} ki te \frac{45}{256} 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.

$$x^{2}+\frac{3\sqrt{5}}{8}x+\frac{45}{256}=-\frac{87}{256}$$

Tauwehea x^{2}+\frac{3\sqrt{5}}{8}x+\frac{45}{256}. 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}.

$$\left(x+\frac{3\sqrt{5}}{16}\right)^{2}=-\frac{87}{256}$$

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

$$\sqrt{\left(x+\frac{3\sqrt{5}}{16}\right)^{2}}=\sqrt{-\frac{87}{256}}$$

Whakarūnātia.

$$x+\frac{3\sqrt{5}}{16}=\frac{\sqrt{87}i}{16} x+\frac{3\sqrt{5}}{16}=-\frac{\sqrt{87}i}{16}$$

Me tango \frac{3\sqrt{5}}{16} mai i ngā taha e rua o te whārite.

$$x=\frac{-3\sqrt{5}+\sqrt{87}i}{16} x=\frac{-\sqrt{87}i-3\sqrt{5}}{16}$$