Whakaoti i te 3-x/left(x+1right)left(x+2right)=15/1

$$\frac{ 3-x }{ \left( x+1 \right) \left( x+2 \right) } = \frac{ 15 }{ 1 }$$

Whakaoti mō x

$$x=\frac{2\sqrt{31}-23}{15}\approx -0.790964752$$

$$x=\frac{-2\sqrt{31}-23}{15}\approx -2.275701915$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -2,-1 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te \left(x+1\right)\left(x+2\right).

$$3-x=\left(x+1\right)\left(x+2\right)\times 15$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x+1 ki te x+2 ka whakakotahi i ngā kupu rite.

$$3-x=\left(x^{2}+3x+2\right)\times 15$$

Whakamahia te āhuatanga tohatoha hei whakarea te x^{2}+3x+2 ki te 15.

$$3-x=15x^{2}+45x+30$$

Tangohia te 15x^{2} mai i ngā taha e rua.

$$3-x-15x^{2}=45x+30$$

Tangohia te 45x mai i ngā taha e rua.

$$3-x-15x^{2}-45x=30$$

Pahekotia te -x me -45x, ka -46x.

$$3-46x-15x^{2}=30$$

Tangohia te 30 mai i ngā taha e rua.

$$3-46x-15x^{2}-30=0$$

Tangohia te 30 i te 3, ka -27.

$$-27-46x-15x^{2}=0$$

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.

$$-15x^{2}-46x-27=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -15 mō a, -46 mō b, me -27 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-46\right)±\sqrt{\left(-46\right)^{2}-4\left(-15\right)\left(-27\right)}}{2\left(-15\right)}$$

Pūrua -46.

$$x=\frac{-\left(-46\right)±\sqrt{2116-4\left(-15\right)\left(-27\right)}}{2\left(-15\right)}$$

Whakareatia -4 ki te -15.

$$x=\frac{-\left(-46\right)±\sqrt{2116+60\left(-27\right)}}{2\left(-15\right)}$$

Whakareatia 60 ki te -27.

$$x=\frac{-\left(-46\right)±\sqrt{2116-1620}}{2\left(-15\right)}$$

Tāpiri 2116 ki te -1620.

$$x=\frac{-\left(-46\right)±\sqrt{496}}{2\left(-15\right)}$$

Tuhia te pūtakerua o te 496.

$$x=\frac{-\left(-46\right)±4\sqrt{31}}{2\left(-15\right)}$$

Ko te tauaro o -46 ko 46.

$$x=\frac{46±4\sqrt{31}}{2\left(-15\right)}$$

Whakareatia 2 ki te -15.

$$x=\frac{46±4\sqrt{31}}{-30}$$

Nā, me whakaoti te whārite x=\frac{46±4\sqrt{31}}{-30} ina he tāpiri te ±. Tāpiri 46 ki te 4\sqrt{31}.

$$x=\frac{4\sqrt{31}+46}{-30}$$

Whakawehe 46+4\sqrt{31} ki te -30.

$$x=\frac{-2\sqrt{31}-23}{15}$$

Nā, me whakaoti te whārite x=\frac{46±4\sqrt{31}}{-30} ina he tango te ±. Tango 4\sqrt{31} mai i 46.

$$x=\frac{46-4\sqrt{31}}{-30}$$

Whakawehe 46-4\sqrt{31} ki te -30.

$$x=\frac{2\sqrt{31}-23}{15}$$

Kua oti te whārite te whakatau.

$$x=\frac{-2\sqrt{31}-23}{15} x=\frac{2\sqrt{31}-23}{15}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -2,-1 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te \left(x+1\right)\left(x+2\right).

$$3-x=\left(x+1\right)\left(x+2\right)\times 15$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x+1 ki te x+2 ka whakakotahi i ngā kupu rite.

$$3-x=\left(x^{2}+3x+2\right)\times 15$$

Whakamahia te āhuatanga tohatoha hei whakarea te x^{2}+3x+2 ki te 15.

$$3-x=15x^{2}+45x+30$$

Tangohia te 15x^{2} mai i ngā taha e rua.

$$3-x-15x^{2}=45x+30$$

Tangohia te 45x mai i ngā taha e rua.

$$3-x-15x^{2}-45x=30$$

Pahekotia te -x me -45x, ka -46x.

$$3-46x-15x^{2}=30$$

Tangohia te 3 mai i ngā taha e rua.

$$-46x-15x^{2}=30-3$$

Tangohia te 3 i te 30, ka 27.

$$-46x-15x^{2}=27$$

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.

$$-15x^{2}-46x=27$$

Whakawehea ngā taha e rua ki te -15.

$$\frac{-15x^{2}-46x}{-15}=\frac{27}{-15}$$

Mā te whakawehe ki te -15 ka wetekia te whakareanga ki te -15.

$$x^{2}+\left(-\frac{46}{-15}\right)x=\frac{27}{-15}$$

Whakawehe -46 ki te -15.

$$x^{2}+\frac{46}{15}x=\frac{27}{-15}$$

Whakahekea te hautanga \frac{27}{-15} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 3.

$$x^{2}+\frac{46}{15}x=-\frac{9}{5}$$

Whakawehea te \frac{46}{15}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{23}{15}. Nā, tāpiria te pūrua o te \frac{23}{15} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{46}{15}x+\left(\frac{23}{15}\right)^{2}=-\frac{9}{5}+\left(\frac{23}{15}\right)^{2}$$

Pūruatia \frac{23}{15} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{46}{15}x+\frac{529}{225}=-\frac{9}{5}+\frac{529}{225}$$

Tāpiri -\frac{9}{5} ki te \frac{529}{225} 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.

$$x^{2}+\frac{46}{15}x+\frac{529}{225}=\frac{124}{225}$$

Tauwehea x^{2}+\frac{46}{15}x+\frac{529}{225}. 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}.

$$\left(x+\frac{23}{15}\right)^{2}=\frac{124}{225}$$

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

$$\sqrt{\left(x+\frac{23}{15}\right)^{2}}=\sqrt{\frac{124}{225}}$$

Whakarūnātia.

$$x+\frac{23}{15}=\frac{2\sqrt{31}}{15} x+\frac{23}{15}=-\frac{2\sqrt{31}}{15}$$

Me tango \frac{23}{15} mai i ngā taha e rua o te whārite.

$$x=\frac{2\sqrt{31}-23}{15} x=\frac{-2\sqrt{31}-23}{15}$$