Whakaoti i te 9(x)=x^2+x+1/x-2

$$9 ( x ) = \frac { x ^ { 2 } + x + 1 } { x - 2 }$$

Whakaoti mō x

$$x = \frac{\sqrt{393} + 19}{16} \approx 2.426514225$$

$$x=\frac{19-\sqrt{393}}{16}\approx -0.051514225$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki 2 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te x-2.

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

Whakamahia te āhuatanga tohatoha hei whakarea te 9x ki te x-2.

$$9x^{2}-18x=x^{2}+x+1$$

Tangohia te x^{2} mai i ngā taha e rua.

$$9x^{2}-18x-x^{2}=x+1$$

Pahekotia te 9x^{2} me -x^{2}, ka 8x^{2}.

$$8x^{2}-18x=x+1$$

Tangohia te x mai i ngā taha e rua.

$$8x^{2}-18x-x=1$$

Pahekotia te -18x me -x, ka -19x.

$$8x^{2}-19x=1$$

Tangohia te 1 mai i ngā taha e rua.

$$8x^{2}-19x-1=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 8 mō a, -19 mō b, me -1 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-19\right)±\sqrt{\left(-19\right)^{2}-4\times 8\left(-1\right)}}{2\times 8}$$

Pūrua -19.

$$x=\frac{-\left(-19\right)±\sqrt{361-4\times 8\left(-1\right)}}{2\times 8}$$

Whakareatia -4 ki te 8.

$$x=\frac{-\left(-19\right)±\sqrt{361-32\left(-1\right)}}{2\times 8}$$

Whakareatia -32 ki te -1.

$$x=\frac{-\left(-19\right)±\sqrt{361+32}}{2\times 8}$$

Tāpiri 361 ki te 32.

$$x=\frac{-\left(-19\right)±\sqrt{393}}{2\times 8}$$

Ko te tauaro o -19 ko 19.

$$x=\frac{19±\sqrt{393}}{2\times 8}$$

Whakareatia 2 ki te 8.

$$x=\frac{19±\sqrt{393}}{16}$$

Nā, me whakaoti te whārite x=\frac{19±\sqrt{393}}{16} ina he tāpiri te ±. Tāpiri 19 ki te \sqrt{393}.

$$x=\frac{\sqrt{393}+19}{16}$$

Nā, me whakaoti te whārite x=\frac{19±\sqrt{393}}{16} ina he tango te ±. Tango \sqrt{393} mai i 19.

$$x=\frac{19-\sqrt{393}}{16}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{393}+19}{16} x=\frac{19-\sqrt{393}}{16}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki 2 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te x-2.

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

Whakamahia te āhuatanga tohatoha hei whakarea te 9x ki te x-2.

$$9x^{2}-18x=x^{2}+x+1$$

Tangohia te x^{2} mai i ngā taha e rua.

$$9x^{2}-18x-x^{2}=x+1$$

Pahekotia te 9x^{2} me -x^{2}, ka 8x^{2}.

$$8x^{2}-18x=x+1$$

Tangohia te x mai i ngā taha e rua.

$$8x^{2}-18x-x=1$$

Pahekotia te -18x me -x, ka -19x.

$$8x^{2}-19x=1$$

Whakawehea ngā taha e rua ki te 8.

$$\frac{8x^{2}-19x}{8}=\frac{1}{8}$$

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

$$x^{2}-\frac{19}{8}x=\frac{1}{8}$$

Whakawehea te -\frac{19}{8}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{19}{16}. Nā, tāpiria te pūrua o te -\frac{19}{16} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{19}{8}x+\left(-\frac{19}{16}\right)^{2}=\frac{1}{8}+\left(-\frac{19}{16}\right)^{2}$$

Pūruatia -\frac{19}{16} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{19}{8}x+\frac{361}{256}=\frac{1}{8}+\frac{361}{256}$$

Tāpiri \frac{1}{8} ki te \frac{361}{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.

$$x^{2}-\frac{19}{8}x+\frac{361}{256}=\frac{393}{256}$$

Tauwehea x^{2}-\frac{19}{8}x+\frac{361}{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}.

$$\left(x-\frac{19}{16}\right)^{2}=\frac{393}{256}$$

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

$$\sqrt{\left(x-\frac{19}{16}\right)^{2}}=\sqrt{\frac{393}{256}}$$

Whakarūnātia.

$$x-\frac{19}{16}=\frac{\sqrt{393}}{16} x-\frac{19}{16}=-\frac{\sqrt{393}}{16}$$

Me tāpiri \frac{19}{16} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{393}+19}{16} x=\frac{19-\sqrt{393}}{16}$$