Whakaoti i te 24/18-x-24/18+x=1

$$\frac { 24 } { 18 - x } - \frac { 24 } { 18 + x } = 1$$

Whakaoti mō x

$$x=-54$$

$$x=6$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -18,18 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(x-18\right)\left(x+18\right), arā, te tauraro pātahi he tino iti rawa te kitea o 18-x,18+x.

$$-\left(18+x\right)\times 24-\left(x-18\right)\times 24=\left(x-18\right)\left(x+18\right)$$

Hei kimi i te tauaro o 18+x, kimihia te tauaro o ia taurangi.

$$\left(-18-x\right)\times 24-\left(x-18\right)\times 24=\left(x-18\right)\left(x+18\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -18-x ki te 24.

$$-432-24x-\left(x-18\right)\times 24=\left(x-18\right)\left(x+18\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-18 ki te 24.

$$-432-24x-\left(24x-432\right)=\left(x-18\right)\left(x+18\right)$$

Hei kimi i te tauaro o 24x-432, kimihia te tauaro o ia taurangi.

$$-432-24x-24x+432=\left(x-18\right)\left(x+18\right)$$

Pahekotia te -24x me -24x, ka -48x.

$$-432-48x+432=\left(x-18\right)\left(x+18\right)$$

Tāpirihia te -432 ki te 432, ka 0.

$$-48x=\left(x-18\right)\left(x+18\right)$$

Whakaarohia te \left(x-18\right)\left(x+18\right). Ka taea te whakareanga te panoni ki te rerekētanga o ngā pūrua mā te ture: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Pūrua 18.

$$-48x=x^{2}-324$$

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

$$-48x-x^{2}=-324$$

Me tāpiri te 324 ki ngā taha e rua.

$$-48x-x^{2}+324=0$$

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.

$$-x^{2}-48x+324=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, -48 mō b, me 324 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-48\right)±\sqrt{\left(-48\right)^{2}-4\left(-1\right)\times 324}}{2\left(-1\right)}$$

Pūrua -48.

$$x=\frac{-\left(-48\right)±\sqrt{2304-4\left(-1\right)\times 324}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-\left(-48\right)±\sqrt{2304+4\times 324}}{2\left(-1\right)}$$

Whakareatia 4 ki te 324.

$$x=\frac{-\left(-48\right)±\sqrt{2304+1296}}{2\left(-1\right)}$$

Tāpiri 2304 ki te 1296.

$$x=\frac{-\left(-48\right)±\sqrt{3600}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 3600.

$$x=\frac{-\left(-48\right)±60}{2\left(-1\right)}$$

Ko te tauaro o -48 ko 48.

$$x=\frac{48±60}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{48±60}{-2}$$

Nā, me whakaoti te whārite x=\frac{48±60}{-2} ina he tāpiri te ±. Tāpiri 48 ki te 60.

$$x=\frac{108}{-2}$$

Whakawehe 108 ki te -2.

$$x=-54$$

Nā, me whakaoti te whārite x=\frac{48±60}{-2} ina he tango te ±. Tango 60 mai i 48.

$$x=-\frac{12}{-2}$$

Whakawehe -12 ki te -2.

$$x=6$$

Kua oti te whārite te whakatau.

$$x=-54 x=6$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -18,18 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(x-18\right)\left(x+18\right), arā, te tauraro pātahi he tino iti rawa te kitea o 18-x,18+x.

$$-\left(18+x\right)\times 24-\left(x-18\right)\times 24=\left(x-18\right)\left(x+18\right)$$

Hei kimi i te tauaro o 18+x, kimihia te tauaro o ia taurangi.

$$\left(-18-x\right)\times 24-\left(x-18\right)\times 24=\left(x-18\right)\left(x+18\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -18-x ki te 24.

$$-432-24x-\left(x-18\right)\times 24=\left(x-18\right)\left(x+18\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-18 ki te 24.

$$-432-24x-\left(24x-432\right)=\left(x-18\right)\left(x+18\right)$$

Hei kimi i te tauaro o 24x-432, kimihia te tauaro o ia taurangi.

$$-432-24x-24x+432=\left(x-18\right)\left(x+18\right)$$

Pahekotia te -24x me -24x, ka -48x.

$$-432-48x+432=\left(x-18\right)\left(x+18\right)$$

Tāpirihia te -432 ki te 432, ka 0.

$$-48x=\left(x-18\right)\left(x+18\right)$$

Whakaarohia te \left(x-18\right)\left(x+18\right). Ka taea te whakareanga te panoni ki te rerekētanga o ngā pūrua mā te ture: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Pūrua 18.

$$-48x=x^{2}-324$$

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

$$-48x-x^{2}=-324$$

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.

$$-x^{2}-48x=-324$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}-48x}{-1}=-\frac{324}{-1}$$

Mā te whakawehe ki te -1 ka wetekia te whakareanga ki te -1.

$$x^{2}+\left(-\frac{48}{-1}\right)x=-\frac{324}{-1}$$

Whakawehe -48 ki te -1.

$$x^{2}+48x=-\frac{324}{-1}$$

Whakawehe -324 ki te -1.

$$x^{2}+48x=324$$

Whakawehea te 48, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 24. Nā, tāpiria te pūrua o te 24 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+48x+24^{2}=324+24^{2}$$

Pūrua 24.

$$x^{2}+48x+576=324+576$$

Tāpiri 324 ki te 576.

$$x^{2}+48x+576=900$$

Tauwehea x^{2}+48x+576. 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}.

$$\left(x+24\right)^{2}=900$$

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

$$\sqrt{\left(x+24\right)^{2}}=\sqrt{900}$$

Whakarūnātia.

$$x+24=30 x+24=-30$$

Me tango 24 mai i ngā taha e rua o te whārite.

$$x=6 x=-54$$