Whakaoti i te x+3/x-3+x-6/x+6=11

$$\frac { x + 3 } { x - 3 } + \frac { x - 6 } { x + 6 } = 11$$

Whakaoti mō x

$$x = \frac{\sqrt{1057} - 11}{6} \approx 3.585256069$$

$$x=\frac{-\sqrt{1057}-11}{6}\approx -7.251922736$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -6,3 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(x-3\right)\left(x+6\right), arā, te tauraro pātahi he tino iti rawa te kitea o x-3,x+6.

$$\left(x+6\right)\left(x+3\right)+\left(x-3\right)\left(x-6\right)=11\left(x-3\right)\left(x+6\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x+6 ki te x+3 ka whakakotahi i ngā kupu rite.

$$x^{2}+9x+18+\left(x-3\right)\left(x-6\right)=11\left(x-3\right)\left(x+6\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x-3 ki te x-6 ka whakakotahi i ngā kupu rite.

$$x^{2}+9x+18+x^{2}-9x+18=11\left(x-3\right)\left(x+6\right)$$

Pahekotia te x^{2} me x^{2}, ka 2x^{2}.

$$2x^{2}+9x+18-9x+18=11\left(x-3\right)\left(x+6\right)$$

Pahekotia te 9x me -9x, ka 0.

$$2x^{2}+18+18=11\left(x-3\right)\left(x+6\right)$$

Tāpirihia te 18 ki te 18, ka 36.

$$2x^{2}+36=11\left(x-3\right)\left(x+6\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 11 ki te x-3.

$$2x^{2}+36=\left(11x-33\right)\left(x+6\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 11x-33 ki te x+6 ka whakakotahi i ngā kupu rite.

$$2x^{2}+36=11x^{2}+33x-198$$

Tangohia te 11x^{2} mai i ngā taha e rua.

$$2x^{2}+36-11x^{2}=33x-198$$

Pahekotia te 2x^{2} me -11x^{2}, ka -9x^{2}.

$$-9x^{2}+36=33x-198$$

Tangohia te 33x mai i ngā taha e rua.

$$-9x^{2}+36-33x=-198$$

Me tāpiri te 198 ki ngā taha e rua.

$$-9x^{2}+36-33x+198=0$$

Tāpirihia te 36 ki te 198, ka 234.

$$-9x^{2}+234-33x=0$$

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.

$$-9x^{2}-33x+234=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -9 mō a, -33 mō b, me 234 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-33\right)±\sqrt{\left(-33\right)^{2}-4\left(-9\right)\times 234}}{2\left(-9\right)}$$

Pūrua -33.

$$x=\frac{-\left(-33\right)±\sqrt{1089-4\left(-9\right)\times 234}}{2\left(-9\right)}$$

Whakareatia -4 ki te -9.

$$x=\frac{-\left(-33\right)±\sqrt{1089+36\times 234}}{2\left(-9\right)}$$

Whakareatia 36 ki te 234.

$$x=\frac{-\left(-33\right)±\sqrt{1089+8424}}{2\left(-9\right)}$$

Tāpiri 1089 ki te 8424.

$$x=\frac{-\left(-33\right)±\sqrt{9513}}{2\left(-9\right)}$$

Tuhia te pūtakerua o te 9513.

$$x=\frac{-\left(-33\right)±3\sqrt{1057}}{2\left(-9\right)}$$

Ko te tauaro o -33 ko 33.

$$x=\frac{33±3\sqrt{1057}}{2\left(-9\right)}$$

Whakareatia 2 ki te -9.

$$x=\frac{33±3\sqrt{1057}}{-18}$$

Nā, me whakaoti te whārite x=\frac{33±3\sqrt{1057}}{-18} ina he tāpiri te ±. Tāpiri 33 ki te 3\sqrt{1057}.

$$x=\frac{3\sqrt{1057}+33}{-18}$$

Whakawehe 33+3\sqrt{1057} ki te -18.

$$x=\frac{-\sqrt{1057}-11}{6}$$

Nā, me whakaoti te whārite x=\frac{33±3\sqrt{1057}}{-18} ina he tango te ±. Tango 3\sqrt{1057} mai i 33.

$$x=\frac{33-3\sqrt{1057}}{-18}$$

Whakawehe 33-3\sqrt{1057} ki te -18.

$$x=\frac{\sqrt{1057}-11}{6}$$

Kua oti te whārite te whakatau.

$$x=\frac{-\sqrt{1057}-11}{6} x=\frac{\sqrt{1057}-11}{6}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -6,3 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(x-3\right)\left(x+6\right), arā, te tauraro pātahi he tino iti rawa te kitea o x-3,x+6.

$$\left(x+6\right)\left(x+3\right)+\left(x-3\right)\left(x-6\right)=11\left(x-3\right)\left(x+6\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x+6 ki te x+3 ka whakakotahi i ngā kupu rite.

$$x^{2}+9x+18+\left(x-3\right)\left(x-6\right)=11\left(x-3\right)\left(x+6\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x-3 ki te x-6 ka whakakotahi i ngā kupu rite.

$$x^{2}+9x+18+x^{2}-9x+18=11\left(x-3\right)\left(x+6\right)$$

Pahekotia te x^{2} me x^{2}, ka 2x^{2}.

$$2x^{2}+9x+18-9x+18=11\left(x-3\right)\left(x+6\right)$$

Pahekotia te 9x me -9x, ka 0.

$$2x^{2}+18+18=11\left(x-3\right)\left(x+6\right)$$

Tāpirihia te 18 ki te 18, ka 36.

$$2x^{2}+36=11\left(x-3\right)\left(x+6\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 11 ki te x-3.

$$2x^{2}+36=\left(11x-33\right)\left(x+6\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 11x-33 ki te x+6 ka whakakotahi i ngā kupu rite.

$$2x^{2}+36=11x^{2}+33x-198$$

Tangohia te 11x^{2} mai i ngā taha e rua.

$$2x^{2}+36-11x^{2}=33x-198$$

Pahekotia te 2x^{2} me -11x^{2}, ka -9x^{2}.

$$-9x^{2}+36=33x-198$$

Tangohia te 33x mai i ngā taha e rua.

$$-9x^{2}+36-33x=-198$$

Tangohia te 36 mai i ngā taha e rua.

$$-9x^{2}-33x=-198-36$$

Tangohia te 36 i te -198, ka -234.

$$-9x^{2}-33x=-234$$

Whakawehea ngā taha e rua ki te -9.

$$\frac{-9x^{2}-33x}{-9}=-\frac{234}{-9}$$

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

$$x^{2}+\left(-\frac{33}{-9}\right)x=-\frac{234}{-9}$$

Whakahekea te hautanga \frac{-33}{-9} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 3.

$$x^{2}+\frac{11}{3}x=-\frac{234}{-9}$$

Whakawehe -234 ki te -9.

$$x^{2}+\frac{11}{3}x=26$$

Whakawehea te \frac{11}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{11}{6}. Nā, tāpiria te pūrua o te \frac{11}{6} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{11}{3}x+\left(\frac{11}{6}\right)^{2}=26+\left(\frac{11}{6}\right)^{2}$$

Pūruatia \frac{11}{6} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{11}{3}x+\frac{121}{36}=26+\frac{121}{36}$$

Tāpiri 26 ki te \frac{121}{36}.

$$x^{2}+\frac{11}{3}x+\frac{121}{36}=\frac{1057}{36}$$

Tauwehea x^{2}+\frac{11}{3}x+\frac{121}{36}. 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}.

$$\left(x+\frac{11}{6}\right)^{2}=\frac{1057}{36}$$

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

$$\sqrt{\left(x+\frac{11}{6}\right)^{2}}=\sqrt{\frac{1057}{36}}$$

Whakarūnātia.

$$x+\frac{11}{6}=\frac{\sqrt{1057}}{6} x+\frac{11}{6}=-\frac{\sqrt{1057}}{6}$$

Me tango \frac{11}{6} mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{1057}-11}{6} x=\frac{-\sqrt{1057}-11}{6}$$