Whakaoti i te 60/x+10+60/x-10=8

$$\frac { 60 } { x + 10 } + \frac { 60 } { x - 10 } = 8$$

Whakaoti mō x

$$x=-5$$

$$x=20$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -10,10 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(x-10\right)\left(x+10\right), arā, te tauraro pātahi he tino iti rawa te kitea o x+10,x-10.

$$\left(x-10\right)\times 60+\left(x+10\right)\times 60=8\left(x-10\right)\left(x+10\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-10 ki te 60.

$$60x-600+\left(x+10\right)\times 60=8\left(x-10\right)\left(x+10\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x+10 ki te 60.

$$60x-600+60x+600=8\left(x-10\right)\left(x+10\right)$$

Pahekotia te 60x me 60x, ka 120x.

$$120x-600+600=8\left(x-10\right)\left(x+10\right)$$

Tāpirihia te -600 ki te 600, ka 0.

$$120x=8\left(x-10\right)\left(x+10\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 8 ki te x-10.

$$120x=\left(8x-80\right)\left(x+10\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 8x-80 ki te x+10 ka whakakotahi i ngā kupu rite.

$$120x=8x^{2}-800$$

Tangohia te 8x^{2} mai i ngā taha e rua.

$$120x-8x^{2}=-800$$

Me tāpiri te 800 ki ngā taha e rua.

$$120x-8x^{2}+800=0$$

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.

$$-8x^{2}+120x+800=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -8 mō a, 120 mō b, me 800 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-120±\sqrt{120^{2}-4\left(-8\right)\times 800}}{2\left(-8\right)}$$

Pūrua 120.

$$x=\frac{-120±\sqrt{14400-4\left(-8\right)\times 800}}{2\left(-8\right)}$$

Whakareatia -4 ki te -8.

$$x=\frac{-120±\sqrt{14400+32\times 800}}{2\left(-8\right)}$$

Whakareatia 32 ki te 800.

$$x=\frac{-120±\sqrt{14400+25600}}{2\left(-8\right)}$$

Tāpiri 14400 ki te 25600.

$$x=\frac{-120±\sqrt{40000}}{2\left(-8\right)}$$

Tuhia te pūtakerua o te 40000.

$$x=\frac{-120±200}{2\left(-8\right)}$$

Whakareatia 2 ki te -8.

$$x=\frac{-120±200}{-16}$$

Nā, me whakaoti te whārite x=\frac{-120±200}{-16} ina he tāpiri te ±. Tāpiri -120 ki te 200.

$$x=\frac{80}{-16}$$

Whakawehe 80 ki te -16.

$$x=-5$$

Nā, me whakaoti te whārite x=\frac{-120±200}{-16} ina he tango te ±. Tango 200 mai i -120.

$$x=-\frac{320}{-16}$$

Whakawehe -320 ki te -16.

$$x=20$$

Kua oti te whārite te whakatau.

$$x=-5 x=20$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -10,10 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(x-10\right)\left(x+10\right), arā, te tauraro pātahi he tino iti rawa te kitea o x+10,x-10.

$$\left(x-10\right)\times 60+\left(x+10\right)\times 60=8\left(x-10\right)\left(x+10\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-10 ki te 60.

$$60x-600+\left(x+10\right)\times 60=8\left(x-10\right)\left(x+10\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x+10 ki te 60.

$$60x-600+60x+600=8\left(x-10\right)\left(x+10\right)$$

Pahekotia te 60x me 60x, ka 120x.

$$120x-600+600=8\left(x-10\right)\left(x+10\right)$$

Tāpirihia te -600 ki te 600, ka 0.

$$120x=8\left(x-10\right)\left(x+10\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 8 ki te x-10.

$$120x=\left(8x-80\right)\left(x+10\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 8x-80 ki te x+10 ka whakakotahi i ngā kupu rite.

$$120x=8x^{2}-800$$

Tangohia te 8x^{2} mai i ngā taha e rua.

$$120x-8x^{2}=-800$$

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.

$$-8x^{2}+120x=-800$$

Whakawehea ngā taha e rua ki te -8.

$$\frac{-8x^{2}+120x}{-8}=-\frac{800}{-8}$$

Mā te whakawehe ki te -8 ka wetekia te whakareanga ki te -8.

$$x^{2}+\frac{120}{-8}x=-\frac{800}{-8}$$

Whakawehe 120 ki te -8.

$$x^{2}-15x=-\frac{800}{-8}$$

Whakawehe -800 ki te -8.

$$x^{2}-15x=100$$

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

$$x^{2}-15x+\left(-\frac{15}{2}\right)^{2}=100+\left(-\frac{15}{2}\right)^{2}$$

Pūruatia -\frac{15}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-15x+\frac{225}{4}=100+\frac{225}{4}$$

Tāpiri 100 ki te \frac{225}{4}.

$$x^{2}-15x+\frac{225}{4}=\frac{625}{4}$$

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

$$\left(x-\frac{15}{2}\right)^{2}=\frac{625}{4}$$

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

$$\sqrt{\left(x-\frac{15}{2}\right)^{2}}=\sqrt{\frac{625}{4}}$$

Whakarūnātia.

$$x-\frac{15}{2}=\frac{25}{2} x-\frac{15}{2}=-\frac{25}{2}$$

Me tāpiri \frac{15}{2} ki ngā taha e rua o te whārite.

$$x=20 x=-5$$