Whakaoti i te 16x^2-64x+65=0

$$16 x ^ { 2 } - 64 x + 65 = 0$$

Whakaoti mō x (complex solution)

$$x=2+\frac{1}{4}i=2+0.25i$$

$$x=2-\frac{1}{4}i=2-0.25i$$

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.

$$16x^{2}-64x+65=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 16 mō a, -64 mō b, me 65 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-64\right)±\sqrt{\left(-64\right)^{2}-4\times 16\times 65}}{2\times 16}$$

Pūrua -64.

$$x=\frac{-\left(-64\right)±\sqrt{4096-4\times 16\times 65}}{2\times 16}$$

Whakareatia -4 ki te 16.

$$x=\frac{-\left(-64\right)±\sqrt{4096-64\times 65}}{2\times 16}$$

Whakareatia -64 ki te 65.

$$x=\frac{-\left(-64\right)±\sqrt{4096-4160}}{2\times 16}$$

Tāpiri 4096 ki te -4160.

$$x=\frac{-\left(-64\right)±\sqrt{-64}}{2\times 16}$$

Tuhia te pūtakerua o te -64.

$$x=\frac{-\left(-64\right)±8i}{2\times 16}$$

Ko te tauaro o -64 ko 64.

$$x=\frac{64±8i}{2\times 16}$$

Whakareatia 2 ki te 16.

$$x=\frac{64±8i}{32}$$

Nā, me whakaoti te whārite x=\frac{64±8i}{32} ina he tāpiri te ±. Tāpiri 64 ki te 8i.

$$x=\frac{64+8i}{32}$$

Whakawehe 64+8i ki te 32.

$$x=2+\frac{1}{4}i$$

Nā, me whakaoti te whārite x=\frac{64±8i}{32} ina he tango te ±. Tango 8i mai i 64.

$$x=\frac{64-8i}{32}$$

Whakawehe 64-8i ki te 32.

$$x=2-\frac{1}{4}i$$

Kua oti te whārite te whakatau.

$$x=2+\frac{1}{4}i x=2-\frac{1}{4}i$$

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.

$$16x^{2}-64x+65=0$$

Me tango 65 mai i ngā taha e rua o te whārite.

$$16x^{2}-64x+65-65=-65$$

Mā te tango i te 65 i a ia ake anō ka toe ko te 0.

$$16x^{2}-64x=-65$$

Whakawehea ngā taha e rua ki te 16.

$$\frac{16x^{2}-64x}{16}=-\frac{65}{16}$$

Mā te whakawehe ki te 16 ka wetekia te whakareanga ki te 16.

$$x^{2}+\left(-\frac{64}{16}\right)x=-\frac{65}{16}$$

Whakawehe -64 ki te 16.

$$x^{2}-4x=-\frac{65}{16}$$

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

$$x^{2}-4x+\left(-2\right)^{2}=-\frac{65}{16}+\left(-2\right)^{2}$$

Pūrua -2.

$$x^{2}-4x+4=-\frac{65}{16}+4$$

Tāpiri -\frac{65}{16} ki te 4.

$$x^{2}-4x+4=-\frac{1}{16}$$

Tauwehea x^{2}-4x+4. 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}.

$$\left(x-2\right)^{2}=-\frac{1}{16}$$

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

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{-\frac{1}{16}}$$

Whakarūnātia.

$$x-2=\frac{1}{4}i x-2=-\frac{1}{4}i$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$x=2+\frac{1}{4}i x=2-\frac{1}{4}i$$