Whakaoti i te 1/x+4/x+1+1=15/x

$$\frac { 1 } { x } + \frac { 4 } { x + 1 } + 1 = \frac { 15 } { x }$$

Whakaoti mō x

$$x = \frac{\sqrt{137} + 9}{2} \approx 10.352349955$$

$$x=\frac{9-\sqrt{137}}{2}\approx -1.352349955$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -1,0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te x\left(x+1\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,x+1.

$$x+1+x\times 4+x\left(x+1\right)=\left(x+1\right)\times 15$$

Pahekotia te x me x\times 4, ka 5x.

$$5x+1+x\left(x+1\right)=\left(x+1\right)\times 15$$

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te x+1.

$$5x+1+x^{2}+x=\left(x+1\right)\times 15$$

Pahekotia te 5x me x, ka 6x.

$$6x+1+x^{2}=\left(x+1\right)\times 15$$

Whakamahia te āhuatanga tohatoha hei whakarea te x+1 ki te 15.

$$6x+1+x^{2}=15x+15$$

Tangohia te 15x mai i ngā taha e rua.

$$6x+1+x^{2}-15x=15$$

Pahekotia te 6x me -15x, ka -9x.

$$-9x+1+x^{2}=15$$

Tangohia te 15 mai i ngā taha e rua.

$$-9x+1+x^{2}-15=0$$

Tangohia te 15 i te 1, ka -14.

$$-9x-14+x^{2}=0$$

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.

$$x^{2}-9x-14=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -9 mō b, me -14 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\left(-14\right)}}{2}$$

Pūrua -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\left(-14\right)}}{2}$$

Whakareatia -4 ki te -14.

$$x=\frac{-\left(-9\right)±\sqrt{81+56}}{2}$$

Tāpiri 81 ki te 56.

$$x=\frac{-\left(-9\right)±\sqrt{137}}{2}$$

Ko te tauaro o -9 ko 9.

$$x=\frac{9±\sqrt{137}}{2}$$

Nā, me whakaoti te whārite x=\frac{9±\sqrt{137}}{2} ina he tāpiri te ±. Tāpiri 9 ki te \sqrt{137}.

$$x=\frac{\sqrt{137}+9}{2}$$

Nā, me whakaoti te whārite x=\frac{9±\sqrt{137}}{2} ina he tango te ±. Tango \sqrt{137} mai i 9.

$$x=\frac{9-\sqrt{137}}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{137}+9}{2} x=\frac{9-\sqrt{137}}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -1,0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te x\left(x+1\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,x+1.

$$x+1+x\times 4+x\left(x+1\right)=\left(x+1\right)\times 15$$

Pahekotia te x me x\times 4, ka 5x.

$$5x+1+x\left(x+1\right)=\left(x+1\right)\times 15$$

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te x+1.

$$5x+1+x^{2}+x=\left(x+1\right)\times 15$$

Pahekotia te 5x me x, ka 6x.

$$6x+1+x^{2}=\left(x+1\right)\times 15$$

Whakamahia te āhuatanga tohatoha hei whakarea te x+1 ki te 15.

$$6x+1+x^{2}=15x+15$$

Tangohia te 15x mai i ngā taha e rua.

$$6x+1+x^{2}-15x=15$$

Pahekotia te 6x me -15x, ka -9x.

$$-9x+1+x^{2}=15$$

Tangohia te 1 mai i ngā taha e rua.

$$-9x+x^{2}=15-1$$

Tangohia te 1 i te 15, ka 14.

$$-9x+x^{2}=14$$

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.

$$x^{2}-9x=14$$

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

$$x^{2}-9x+\left(-\frac{9}{2}\right)^{2}=14+\left(-\frac{9}{2}\right)^{2}$$

Pūruatia -\frac{9}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-9x+\frac{81}{4}=14+\frac{81}{4}$$

Tāpiri 14 ki te \frac{81}{4}.

$$x^{2}-9x+\frac{81}{4}=\frac{137}{4}$$

Tauwehea x^{2}-9x+\frac{81}{4}. 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}.

$$\left(x-\frac{9}{2}\right)^{2}=\frac{137}{4}$$

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

$$\sqrt{\left(x-\frac{9}{2}\right)^{2}}=\sqrt{\frac{137}{4}}$$

Whakarūnātia.

$$x-\frac{9}{2}=\frac{\sqrt{137}}{2} x-\frac{9}{2}=-\frac{\sqrt{137}}{2}$$

Me tāpiri \frac{9}{2} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{137}+9}{2} x=\frac{9-\sqrt{137}}{2}$$