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

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

Whakaoti mō x (complex solution)

$$x=-5+5\sqrt{287}i\approx -5+84.70537173i$$

$$x=-5\sqrt{287}i-5\approx -5-84.70537173i$$

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 tango rāua mā te tango i ō raua taurunga.

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

Mahia ngā whakarea i roto o x-\left(x+10\right).

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

Whakakotahitia ngā kupu rite i x-x-10.

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

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{-10}{x\left(x+10\right)} mā te whakarea 1 ki te tau huripoki o \frac{-10}{x\left(x+10\right)}.

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

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

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

Whakawehea ia wā o x^{2}+10x ki te -10, kia riro ko -\frac{1}{10}x^{2}-x.

$$-\frac{1}{10}x^{2}-x=720$$

Tangohia te 720 mai i ngā taha e rua.

$$-\frac{1}{10}x^{2}-x-720=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -\frac{1}{10} mō a, -1 mō b, me -720 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\left(-\frac{1}{10}\right)\left(-720\right)}}{2\left(-\frac{1}{10}\right)}$$

Whakareatia -4 ki te -\frac{1}{10}.

$$x=\frac{-\left(-1\right)±\sqrt{1+\frac{2}{5}\left(-720\right)}}{2\left(-\frac{1}{10}\right)}$$

Whakareatia \frac{2}{5} ki te -720.

$$x=\frac{-\left(-1\right)±\sqrt{1-288}}{2\left(-\frac{1}{10}\right)}$$

Tāpiri 1 ki te -288.

$$x=\frac{-\left(-1\right)±\sqrt{-287}}{2\left(-\frac{1}{10}\right)}$$

Tuhia te pūtakerua o te -287.

$$x=\frac{-\left(-1\right)±\sqrt{287}i}{2\left(-\frac{1}{10}\right)}$$

Ko te tauaro o -1 ko 1.

$$x=\frac{1±\sqrt{287}i}{2\left(-\frac{1}{10}\right)}$$

Whakareatia 2 ki te -\frac{1}{10}.

$$x=\frac{1±\sqrt{287}i}{-\frac{1}{5}}$$

Nā, me whakaoti te whārite x=\frac{1±\sqrt{287}i}{-\frac{1}{5}} ina he tāpiri te ±. Tāpiri 1 ki te i\sqrt{287}.

$$x=\frac{1+\sqrt{287}i}{-\frac{1}{5}}$$

Whakawehe 1+i\sqrt{287} ki te -\frac{1}{5} mā te whakarea 1+i\sqrt{287} ki te tau huripoki o -\frac{1}{5}.

$$x=-5\sqrt{287}i-5$$

Nā, me whakaoti te whārite x=\frac{1±\sqrt{287}i}{-\frac{1}{5}} ina he tango te ±. Tango i\sqrt{287} mai i 1.

$$x=\frac{-\sqrt{287}i+1}{-\frac{1}{5}}$$

Whakawehe 1-i\sqrt{287} ki te -\frac{1}{5} mā te whakarea 1-i\sqrt{287} ki te tau huripoki o -\frac{1}{5}.

$$x=-5+5\sqrt{287}i$$

Kua oti te whārite te whakatau.

$$x=-5\sqrt{287}i-5 x=-5+5\sqrt{287}i$$

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 tango rāua mā te tango i ō raua taurunga.

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

Mahia ngā whakarea i roto o x-\left(x+10\right).

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

Whakakotahitia ngā kupu rite i x-x-10.

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

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{-10}{x\left(x+10\right)} mā te whakarea 1 ki te tau huripoki o \frac{-10}{x\left(x+10\right)}.

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

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

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

Whakawehea ia wā o x^{2}+10x ki te -10, kia riro ko -\frac{1}{10}x^{2}-x.

$$-\frac{1}{10}x^{2}-x=720$$

Me whakarea ngā taha e rua ki te -10.

$$\frac{-\frac{1}{10}x^{2}-x}{-\frac{1}{10}}=\frac{720}{-\frac{1}{10}}$$

Mā te whakawehe ki te -\frac{1}{10} ka wetekia te whakareanga ki te -\frac{1}{10}.

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

Whakawehe -1 ki te -\frac{1}{10} mā te whakarea -1 ki te tau huripoki o -\frac{1}{10}.

$$x^{2}+10x=\frac{720}{-\frac{1}{10}}$$

Whakawehe 720 ki te -\frac{1}{10} mā te whakarea 720 ki te tau huripoki o -\frac{1}{10}.

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

Whakawehea te 10, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 5. Nā, tāpiria te pūrua o te 5 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+10x+5^{2}=-7200+5^{2}$$

Pūrua 5.

$$x^{2}+10x+25=-7200+25$$

Tāpiri -7200 ki te 25.

$$x^{2}+10x+25=-7175$$

Tauwehea x^{2}+10x+25. 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}.

$$\left(x+5\right)^{2}=-7175$$

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

$$\sqrt{\left(x+5\right)^{2}}=\sqrt{-7175}$$

Whakarūnātia.

$$x+5=5\sqrt{287}i x+5=-5\sqrt{287}i$$

Me tango 5 mai i ngā taha e rua o te whārite.

$$x=-5+5\sqrt{287}i x=-5\sqrt{287}i-5$$