Whakaoti i te 8x^2-8x-1=0

$$8 x ^ { 2 } - 8 x - 1 = 0$$

Whakaoti mō x

$$x=\frac{\sqrt{6}}{4}+\frac{1}{2}\approx 1.112372436$$

$$x=-\frac{\sqrt{6}}{4}+\frac{1}{2}\approx -0.112372436$$

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.

$$8x^{2}-8x-1=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 8 mō a, -8 mō b, me -1 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 8\left(-1\right)}}{2\times 8}$$

Pūrua -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\times 8\left(-1\right)}}{2\times 8}$$

Whakareatia -4 ki te 8.

$$x=\frac{-\left(-8\right)±\sqrt{64-32\left(-1\right)}}{2\times 8}$$

Whakareatia -32 ki te -1.

$$x=\frac{-\left(-8\right)±\sqrt{64+32}}{2\times 8}$$

Tāpiri 64 ki te 32.

$$x=\frac{-\left(-8\right)±\sqrt{96}}{2\times 8}$$

Tuhia te pūtakerua o te 96.

$$x=\frac{-\left(-8\right)±4\sqrt{6}}{2\times 8}$$

Ko te tauaro o -8 ko 8.

$$x=\frac{8±4\sqrt{6}}{2\times 8}$$

Whakareatia 2 ki te 8.

$$x=\frac{8±4\sqrt{6}}{16}$$

Nā, me whakaoti te whārite x=\frac{8±4\sqrt{6}}{16} ina he tāpiri te ±. Tāpiri 8 ki te 4\sqrt{6}.

$$x=\frac{4\sqrt{6}+8}{16}$$

Whakawehe 8+4\sqrt{6} ki te 16.

$$x=\frac{\sqrt{6}}{4}+\frac{1}{2}$$

Nā, me whakaoti te whārite x=\frac{8±4\sqrt{6}}{16} ina he tango te ±. Tango 4\sqrt{6} mai i 8.

$$x=\frac{8-4\sqrt{6}}{16}$$

Whakawehe 8-4\sqrt{6} ki te 16.

$$x=-\frac{\sqrt{6}}{4}+\frac{1}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{6}}{4}+\frac{1}{2} x=-\frac{\sqrt{6}}{4}+\frac{1}{2}$$

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.

$$8x^{2}-8x-1=0$$

Me tāpiri 1 ki ngā taha e rua o te whārite.

$$8x^{2}-8x-1-\left(-1\right)=-\left(-1\right)$$

Mā te tango i te -1 i a ia ake anō ka toe ko te 0.

$$8x^{2}-8x=-\left(-1\right)$$

Tango -1 mai i 0.

$$8x^{2}-8x=1$$

Whakawehea ngā taha e rua ki te 8.

$$\frac{8x^{2}-8x}{8}=\frac{1}{8}$$

Mā te whakawehe ki te 8 ka wetekia te whakareanga ki te 8.

$$x^{2}+\left(-\frac{8}{8}\right)x=\frac{1}{8}$$

Whakawehe -8 ki te 8.

$$x^{2}-x=\frac{1}{8}$$

Whakawehea te -1, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{2}. Nā, tāpiria te pūrua o te -\frac{1}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=\frac{1}{8}+\left(-\frac{1}{2}\right)^{2}$$

Pūruatia -\frac{1}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-x+\frac{1}{4}=\frac{1}{8}+\frac{1}{4}$$

Tāpiri \frac{1}{8} ki te \frac{1}{4} 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.

$$x^{2}-x+\frac{1}{4}=\frac{3}{8}$$

Tauwehea te x^{2}-x+\frac{1}{4}. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{3}{8}$$

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

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{3}{8}}$$

Whakarūnātia.

$$x-\frac{1}{2}=\frac{\sqrt{6}}{4} x-\frac{1}{2}=-\frac{\sqrt{6}}{4}$$

Me tāpiri \frac{1}{2} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{6}}{4}+\frac{1}{2} x=-\frac{\sqrt{6}}{4}+\frac{1}{2}$$