Whakaoti i te 9x^2-x-890=0

$$9 x ^ { 2 } - x - 890 = 0$$

Whakaoti mō x

$$x = -\frac{89}{9} = -9\frac{8}{9} \approx -9.888888889$$

$$x=10$$

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 9x^{2}+ax+bx-890. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-1 ab=9\left(-890\right)=-8010$$

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 -8010.

$$1,-8010 2,-4005 3,-2670 5,-1602 6,-1335 9,-890 10,-801 15,-534 18,-445 30,-267 45,-178 89,-90$$

Tātaihia te tapeke mō ia takirua.

$$1-8010=-8009 2-4005=-4003 3-2670=-2667 5-1602=-1597 6-1335=-1329 9-890=-881 10-801=-791 15-534=-519 18-445=-427 30-267=-237 45-178=-133 89-90=-1$$

Ko te otinga te takirua ka hoatu i te tapeke -1.

$$a=-90 b=89$$

Tuhia anō te 9x^{2}-x-890 hei \left(9x^{2}-90x\right)+\left(89x-890\right).

$$\left(9x^{2}-90x\right)+\left(89x-890\right)$$

Tauwehea te 9x i te tuatahi me te 89 i te rōpū tuarua.

$$9x\left(x-10\right)+89\left(x-10\right)$$

Whakatauwehea atu te kīanga pātahi x-10 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-10\right)\left(9x+89\right)$$

Hei kimi otinga whārite, me whakaoti te x-10=0 me te 9x+89=0.

$$x=10 x=-\frac{89}{9}$$

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.

$$9x^{2}-x-890=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 9 mō a, -1 mō b, me -890 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times 9\left(-890\right)}}{2\times 9}$$

Whakareatia -4 ki te 9.

$$x=\frac{-\left(-1\right)±\sqrt{1-36\left(-890\right)}}{2\times 9}$$

Whakareatia -36 ki te -890.

$$x=\frac{-\left(-1\right)±\sqrt{1+32040}}{2\times 9}$$

Tāpiri 1 ki te 32040.

$$x=\frac{-\left(-1\right)±\sqrt{32041}}{2\times 9}$$

Tuhia te pūtakerua o te 32041.

$$x=\frac{-\left(-1\right)±179}{2\times 9}$$

Ko te tauaro o -1 ko 1.

$$x=\frac{1±179}{2\times 9}$$

Whakareatia 2 ki te 9.

$$x=\frac{1±179}{18}$$

Nā, me whakaoti te whārite x=\frac{1±179}{18} ina he tāpiri te ±. Tāpiri 1 ki te 179.

$$x=\frac{180}{18}$$

Whakawehe 180 ki te 18.

$$x=10$$

Nā, me whakaoti te whārite x=\frac{1±179}{18} ina he tango te ±. Tango 179 mai i 1.

$$x=-\frac{178}{18}$$

Whakahekea te hautanga \frac{-178}{18} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=-\frac{89}{9}$$

Kua oti te whārite te whakatau.

$$x=10 x=-\frac{89}{9}$$

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.

$$9x^{2}-x-890=0$$

Me tāpiri 890 ki ngā taha e rua o te whārite.

$$9x^{2}-x-890-\left(-890\right)=-\left(-890\right)$$

Mā te tango i te -890 i a ia ake anō ka toe ko te 0.

$$9x^{2}-x=-\left(-890\right)$$

Tango -890 mai i 0.

$$9x^{2}-x=890$$

Whakawehea ngā taha e rua ki te 9.

$$\frac{9x^{2}-x}{9}=\frac{890}{9}$$

Mā te whakawehe ki te 9 ka wetekia te whakareanga ki te 9.

$$x^{2}-\frac{1}{9}x=\frac{890}{9}$$

Whakawehea te -\frac{1}{9}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{18}. Nā, tāpiria te pūrua o te -\frac{1}{18} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{1}{9}x+\left(-\frac{1}{18}\right)^{2}=\frac{890}{9}+\left(-\frac{1}{18}\right)^{2}$$

Pūruatia -\frac{1}{18} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{1}{9}x+\frac{1}{324}=\frac{890}{9}+\frac{1}{324}$$

Tāpiri \frac{890}{9} ki te \frac{1}{324} 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.

$$x^{2}-\frac{1}{9}x+\frac{1}{324}=\frac{32041}{324}$$

Tauwehea x^{2}-\frac{1}{9}x+\frac{1}{324}. 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}.

$$\left(x-\frac{1}{18}\right)^{2}=\frac{32041}{324}$$

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

$$\sqrt{\left(x-\frac{1}{18}\right)^{2}}=\sqrt{\frac{32041}{324}}$$

Whakarūnātia.

$$x-\frac{1}{18}=\frac{179}{18} x-\frac{1}{18}=-\frac{179}{18}$$

Me tāpiri \frac{1}{18} ki ngā taha e rua o te whārite.

$$x=10 x=-\frac{89}{9}$$