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

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

Whakaoti mō x

$$x=5\sqrt{20737}+725\approx 1445.017360902$$

$$x=725-5\sqrt{20737}\approx 4.982639098$$

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 me x-10 ko x\left(x-10\right). Whakareatia \frac{1}{x} ki te \frac{x-10}{x-10}. Whakareatia \frac{1}{x-10} ki te \frac{x}{x}.

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

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

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

Whakakotahitia ngā kupu rite i x-10+x.

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

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara 0,10 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.

$$\frac{x^{2}-1450x+7200}{2\left(x-5\right)}=0$$

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).

$$x^{2}-1450x+7200=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -1450 mō b, me 7200 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1450\right)±\sqrt{\left(-1450\right)^{2}-4\times 7200}}{2}$$

Pūrua -1450.

$$x=\frac{-\left(-1450\right)±\sqrt{2102500-4\times 7200}}{2}$$

Whakareatia -4 ki te 7200.

$$x=\frac{-\left(-1450\right)±\sqrt{2102500-28800}}{2}$$

Tāpiri 2102500 ki te -28800.

$$x=\frac{-\left(-1450\right)±\sqrt{2073700}}{2}$$

Tuhia te pūtakerua o te 2073700.

$$x=\frac{-\left(-1450\right)±10\sqrt{20737}}{2}$$

Ko te tauaro o -1450 ko 1450.

$$x=\frac{1450±10\sqrt{20737}}{2}$$

Nā, me whakaoti te whārite x=\frac{1450±10\sqrt{20737}}{2} ina he tāpiri te ±. Tāpiri 1450 ki te 10\sqrt{20737}.

$$x=\frac{10\sqrt{20737}+1450}{2}$$

Whakawehe 1450+10\sqrt{20737} ki te 2.

$$x=5\sqrt{20737}+725$$

Nā, me whakaoti te whārite x=\frac{1450±10\sqrt{20737}}{2} ina he tango te ±. Tango 10\sqrt{20737} mai i 1450.

$$x=\frac{1450-10\sqrt{20737}}{2}$$

Whakawehe 1450-10\sqrt{20737} ki te 2.

$$x=725-5\sqrt{20737}$$

Kua oti te whārite te whakatau.

$$x=5\sqrt{20737}+725 x=725-5\sqrt{20737}$$

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 me x-10 ko x\left(x-10\right). Whakareatia \frac{1}{x} ki te \frac{x-10}{x-10}. Whakareatia \frac{1}{x-10} ki te \frac{x}{x}.

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

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

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

Whakakotahitia ngā kupu rite i x-10+x.

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

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara 0,10 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 -1450x.

$$x^{2}-1450x=-7200$$

Whakawehea te -1450, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -725. Nā, tāpiria te pūrua o te -725 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-1450x+\left(-725\right)^{2}=-7200+\left(-725\right)^{2}$$

Pūrua -725.

$$x^{2}-1450x+525625=-7200+525625$$

Tāpiri -7200 ki te 525625.

$$x^{2}-1450x+525625=518425$$

Tauwehea x^{2}-1450x+525625. 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}.

$$\left(x-725\right)^{2}=518425$$

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

$$\sqrt{\left(x-725\right)^{2}}=\sqrt{518425}$$

Whakarūnātia.

$$x-725=5\sqrt{20737} x-725=-5\sqrt{20737}$$

Me tāpiri 725 ki ngā taha e rua o te whārite.

$$x=5\sqrt{20737}+725 x=725-5\sqrt{20737}$$