Whakaoti i te 8x^2-7x-1=0

$$8 x ^ { 2 } - 7 x - 1 = 0$$

Whakaoti mō x

$$x=-\frac{1}{8}=-0.125$$

$$x=1$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 8x^{2}+ax+bx-1. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-7 ab=8\left(-1\right)=-8$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -8.

$$1,-8 2,-4$$

Tātaihia te tapeke mō ia takirua.

$$1-8=-7 2-4=-2$$

Ko te otinga te takirua ka hoatu i te tapeke -7.

$$a=-8 b=1$$

Tuhia anō te 8x^{2}-7x-1 hei \left(8x^{2}-8x\right)+\left(x-1\right).

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

Whakatauwehea atu 8x i te 8x^{2}-8x.

$$8x\left(x-1\right)+x-1$$

Whakatauwehea atu te kīanga pātahi x-1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-1\right)\left(8x+1\right)$$

Hei kimi otinga whārite, me whakaoti te x-1=0 me te 8x+1=0.

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

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.

$$8x^{2}-7x-1=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 8 mō a, -7 mō b, me -1 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 8\left(-1\right)}}{2\times 8}$$

Pūrua -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\times 8\left(-1\right)}}{2\times 8}$$

Whakareatia -4 ki te 8.

$$x=\frac{-\left(-7\right)±\sqrt{49-32\left(-1\right)}}{2\times 8}$$

Whakareatia -32 ki te -1.

$$x=\frac{-\left(-7\right)±\sqrt{49+32}}{2\times 8}$$

Tāpiri 49 ki te 32.

$$x=\frac{-\left(-7\right)±\sqrt{81}}{2\times 8}$$

Tuhia te pūtakerua o te 81.

$$x=\frac{-\left(-7\right)±9}{2\times 8}$$

Ko te tauaro o -7 ko 7.

$$x=\frac{7±9}{2\times 8}$$

Whakareatia 2 ki te 8.

$$x=\frac{7±9}{16}$$

Nā, me whakaoti te whārite x=\frac{7±9}{16} ina he tāpiri te ±. Tāpiri 7 ki te 9.

$$x=\frac{16}{16}$$

Whakawehe 16 ki te 16.

$$x=1$$

Nā, me whakaoti te whārite x=\frac{7±9}{16} ina he tango te ±. Tango 9 mai i 7.

$$x=-\frac{2}{16}$$

Whakahekea te hautanga \frac{-2}{16} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

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

Kua oti te whārite te whakatau.

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

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.

$$8x^{2}-7x-1=0$$

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

$$8x^{2}-7x-1-\left(-1\right)=-\left(-1\right)$$

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

$$8x^{2}-7x=-\left(-1\right)$$

Tango -1 mai i 0.

$$8x^{2}-7x=1$$

Whakawehea ngā taha e rua ki te 8.

$$\frac{8x^{2}-7x}{8}=\frac{1}{8}$$

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

$$x^{2}-\frac{7}{8}x=\frac{1}{8}$$

Whakawehea te -\frac{7}{8}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{7}{16}. Nā, tāpiria te pūrua o te -\frac{7}{16} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{7}{8}x+\left(-\frac{7}{16}\right)^{2}=\frac{1}{8}+\left(-\frac{7}{16}\right)^{2}$$

Pūruatia -\frac{7}{16} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{7}{8}x+\frac{49}{256}=\frac{1}{8}+\frac{49}{256}$$

Tāpiri \frac{1}{8} ki te \frac{49}{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.

$$x^{2}-\frac{7}{8}x+\frac{49}{256}=\frac{81}{256}$$

Tauwehea x^{2}-\frac{7}{8}x+\frac{49}{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}.

$$\left(x-\frac{7}{16}\right)^{2}=\frac{81}{256}$$

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

$$\sqrt{\left(x-\frac{7}{16}\right)^{2}}=\sqrt{\frac{81}{256}}$$

Whakarūnātia.

$$x-\frac{7}{16}=\frac{9}{16} x-\frac{7}{16}=-\frac{9}{16}$$

Me tāpiri \frac{7}{16} ki ngā taha e rua o te whārite.

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