Whakaoti i te 1/frac{1{x+10}+1/x}=720

$$\frac{ 1 }{ \frac{ 1 }{ x+10 } + \frac{ 1 }{ x } } = 720$$

Whakaoti mō x

$$x=5\sqrt{20737}+715\approx 1435.017360902$$

$$x=715-5\sqrt{20737}\approx -5.017360902$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Ko te taurea pātahi iti rawa o x+10 me x ko x\left(x+10\right). Whakareatia \frac{1}{x+10} ki te \frac{x}{x}. Whakareatia \frac{1}{x} ki te \frac{x+10}{x+10}.

$$\frac{1}{\frac{x}{x\left(x+10\right)}+\frac{x+10}{x\left(x+10\right)}}=720$$

Tā te mea he rite te tauraro o \frac{x}{x\left(x+10\right)} me \frac{x+10}{x\left(x+10\right)}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$\frac{1}{\frac{x+x+10}{x\left(x+10\right)}}=720$$

Whakakotahitia ngā kupu rite i x+x+10.

$$\frac{1}{\frac{2x+10}{x\left(x+10\right)}}=720$$

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -10,0 nā te kore tautuhi i te whakawehenga mā te kore. Whakawehe 1 ki te \frac{2x+10}{x\left(x+10\right)} mā te whakarea 1 ki te tau huripoki o \frac{2x+10}{x\left(x+10\right)}.

$$\frac{x\left(x+10\right)}{2x+10}=720$$

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te x+10.

$$\frac{x^{2}+10x}{2x+10}=720$$

Tangohia te 720 mai i ngā taha e rua.

$$\frac{x^{2}+10x}{2x+10}-720=0$$

Tauwehea te 2x+10.

$$\frac{x^{2}+10x}{2\left(x+5\right)}-720=0$$

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Whakareatia 720 ki te \frac{2\left(x+5\right)}{2\left(x+5\right)}.

$$\frac{x^{2}+10x}{2\left(x+5\right)}-\frac{720\times 2\left(x+5\right)}{2\left(x+5\right)}=0$$

Tā te mea he rite te tauraro o \frac{x^{2}+10x}{2\left(x+5\right)} me \frac{720\times 2\left(x+5\right)}{2\left(x+5\right)}, me tango rāua mā te tango i ō raua taurunga.

$$\frac{x^{2}+10x-720\times 2\left(x+5\right)}{2\left(x+5\right)}=0$$

Mahia ngā whakarea i roto o x^{2}+10x-720\times 2\left(x+5\right).

$$\frac{x^{2}+10x-1440x-7200}{2\left(x+5\right)}=0$$

Whakakotahitia ngā kupu rite i x^{2}+10x-1440x-7200.

$$\frac{x^{2}-1430x-7200}{2\left(x+5\right)}=0$$

Tē taea kia ōrite te tāupe x ki -5 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te 2\left(x+5\right).

$$x^{2}-1430x-7200=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -1430 mō b, me -7200 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1430\right)±\sqrt{\left(-1430\right)^{2}-4\left(-7200\right)}}{2}$$

Pūrua -1430.

$$x=\frac{-\left(-1430\right)±\sqrt{2044900-4\left(-7200\right)}}{2}$$

Whakareatia -4 ki te -7200.

$$x=\frac{-\left(-1430\right)±\sqrt{2044900+28800}}{2}$$

Tāpiri 2044900 ki te 28800.

$$x=\frac{-\left(-1430\right)±\sqrt{2073700}}{2}$$

Tuhia te pūtakerua o te 2073700.

$$x=\frac{-\left(-1430\right)±10\sqrt{20737}}{2}$$

Ko te tauaro o -1430 ko 1430.

$$x=\frac{1430±10\sqrt{20737}}{2}$$

Nā, me whakaoti te whārite x=\frac{1430±10\sqrt{20737}}{2} ina he tāpiri te ±. Tāpiri 1430 ki te 10\sqrt{20737}.

$$x=\frac{10\sqrt{20737}+1430}{2}$$

Whakawehe 1430+10\sqrt{20737} ki te 2.

$$x=5\sqrt{20737}+715$$

Nā, me whakaoti te whārite x=\frac{1430±10\sqrt{20737}}{2} ina he tango te ±. Tango 10\sqrt{20737} mai i 1430.

$$x=\frac{1430-10\sqrt{20737}}{2}$$

Whakawehe 1430-10\sqrt{20737} ki te 2.

$$x=715-5\sqrt{20737}$$

Kua oti te whārite te whakatau.

$$x=5\sqrt{20737}+715 x=715-5\sqrt{20737}$$

Ngā Upane mō te Whakaoti i te Pūrua

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Ko te taurea pātahi iti rawa o x+10 me x ko x\left(x+10\right). Whakareatia \frac{1}{x+10} ki te \frac{x}{x}. Whakareatia \frac{1}{x} ki te \frac{x+10}{x+10}.

$$\frac{1}{\frac{x}{x\left(x+10\right)}+\frac{x+10}{x\left(x+10\right)}}=720$$

Tā te mea he rite te tauraro o \frac{x}{x\left(x+10\right)} me \frac{x+10}{x\left(x+10\right)}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$\frac{1}{\frac{x+x+10}{x\left(x+10\right)}}=720$$

Whakakotahitia ngā kupu rite i x+x+10.

$$\frac{1}{\frac{2x+10}{x\left(x+10\right)}}=720$$

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -10,0 nā te kore tautuhi i te whakawehenga mā te kore. Whakawehe 1 ki te \frac{2x+10}{x\left(x+10\right)} mā te whakarea 1 ki te tau huripoki o \frac{2x+10}{x\left(x+10\right)}.

$$\frac{x\left(x+10\right)}{2x+10}=720$$

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te x+10.

$$\frac{x^{2}+10x}{2x+10}=720$$

Tē taea kia ōrite te tāupe x ki -5 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te 2\left(x+5\right).

$$x^{2}+10x=1440\left(x+5\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 1440 ki te x+5.

$$x^{2}+10x=1440x+7200$$

Tangohia te 1440x mai i ngā taha e rua.

$$x^{2}+10x-1440x=7200$$

Pahekotia te 10x me -1440x, ka -1430x.

$$x^{2}-1430x=7200$$

Whakawehea te -1430, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -715. Nā, tāpiria te pūrua o te -715 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-1430x+\left(-715\right)^{2}=7200+\left(-715\right)^{2}$$

Pūrua -715.

$$x^{2}-1430x+511225=7200+511225$$

Tāpiri 7200 ki te 511225.

$$x^{2}-1430x+511225=518425$$

Tauwehea x^{2}-1430x+511225. 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}.

$$\left(x-715\right)^{2}=518425$$

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

$$\sqrt{\left(x-715\right)^{2}}=\sqrt{518425}$$

Whakarūnātia.

$$x-715=5\sqrt{20737} x-715=-5\sqrt{20737}$$

Me tāpiri 715 ki ngā taha e rua o te whārite.

$$x=5\sqrt{20737}+715 x=715-5\sqrt{20737}$$