Whakaoti i te 14*x/12+x*14/12+x=4

$$14 \times \frac{ x }{ 12+x } \times \frac{ 14 }{ 12+x } =4$$

Whakaoti mō x

$$x=9$$

$$x=16$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki -12 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te x+12.

$$14x\times \frac{14}{12+x}=4\left(x+12\right)$$

Tuhia te 14\times \frac{14}{12+x} hei hautanga kotahi.

$$\frac{14\times 14}{12+x}x=4\left(x+12\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te x+12.

$$\frac{14\times 14}{12+x}x=4x+48$$

Whakareatia te 14 ki te 14, ka 196.

$$\frac{196}{12+x}x=4x+48$$

Tuhia te \frac{196}{12+x}x hei hautanga kotahi.

$$\frac{196x}{12+x}=4x+48$$

Tangohia te 4x mai i ngā taha e rua.

$$\frac{196x}{12+x}-4x=48$$

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Whakareatia -4x ki te \frac{12+x}{12+x}.

$$\frac{196x}{12+x}+\frac{-4x\left(12+x\right)}{12+x}=48$$

Tā te mea he rite te tauraro o \frac{196x}{12+x} me \frac{-4x\left(12+x\right)}{12+x}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$\frac{196x-4x\left(12+x\right)}{12+x}=48$$

Mahia ngā whakarea i roto o 196x-4x\left(12+x\right).

$$\frac{196x-48x-4x^{2}}{12+x}=48$$

Whakakotahitia ngā kupu rite i 196x-48x-4x^{2}.

$$\frac{148x-4x^{2}}{12+x}=48$$

Tangohia te 48 mai i ngā taha e rua.

$$\frac{148x-4x^{2}}{12+x}-48=0$$

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Whakareatia 48 ki te \frac{12+x}{12+x}.

$$\frac{148x-4x^{2}}{12+x}-\frac{48\left(12+x\right)}{12+x}=0$$

Tā te mea he rite te tauraro o \frac{148x-4x^{2}}{12+x} me \frac{48\left(12+x\right)}{12+x}, me tango rāua mā te tango i ō raua taurunga.

$$\frac{148x-4x^{2}-48\left(12+x\right)}{12+x}=0$$

Mahia ngā whakarea i roto o 148x-4x^{2}-48\left(12+x\right).

$$\frac{148x-4x^{2}-576-48x}{12+x}=0$$

Whakakotahitia ngā kupu rite i 148x-4x^{2}-576-48x.

$$\frac{100x-4x^{2}-576}{12+x}=0$$

Tē taea kia ōrite te tāupe x ki -12 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te x+12.

$$100x-4x^{2}-576=0$$

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.

$$-4x^{2}+100x-576=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -4 mō a, 100 mō b, me -576 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-100±\sqrt{100^{2}-4\left(-4\right)\left(-576\right)}}{2\left(-4\right)}$$

Pūrua 100.

$$x=\frac{-100±\sqrt{10000-4\left(-4\right)\left(-576\right)}}{2\left(-4\right)}$$

Whakareatia -4 ki te -4.

$$x=\frac{-100±\sqrt{10000+16\left(-576\right)}}{2\left(-4\right)}$$

Whakareatia 16 ki te -576.

$$x=\frac{-100±\sqrt{10000-9216}}{2\left(-4\right)}$$

Tāpiri 10000 ki te -9216.

$$x=\frac{-100±\sqrt{784}}{2\left(-4\right)}$$

Tuhia te pūtakerua o te 784.

$$x=\frac{-100±28}{2\left(-4\right)}$$

Whakareatia 2 ki te -4.

$$x=\frac{-100±28}{-8}$$

Nā, me whakaoti te whārite x=\frac{-100±28}{-8} ina he tāpiri te ±. Tāpiri -100 ki te 28.

$$x=-\frac{72}{-8}$$

Whakawehe -72 ki te -8.

$$x=9$$

Nā, me whakaoti te whārite x=\frac{-100±28}{-8} ina he tango te ±. Tango 28 mai i -100.

$$x=-\frac{128}{-8}$$

Whakawehe -128 ki te -8.

$$x=16$$

Kua oti te whārite te whakatau.

$$x=9 x=16$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki -12 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te x+12.

$$14x\times \frac{14}{12+x}=4\left(x+12\right)$$

Tuhia te 14\times \frac{14}{12+x} hei hautanga kotahi.

$$\frac{14\times 14}{12+x}x=4\left(x+12\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te x+12.

$$\frac{14\times 14}{12+x}x=4x+48$$

Whakareatia te 14 ki te 14, ka 196.

$$\frac{196}{12+x}x=4x+48$$

Tuhia te \frac{196}{12+x}x hei hautanga kotahi.

$$\frac{196x}{12+x}=4x+48$$

Tangohia te 4x mai i ngā taha e rua.

$$\frac{196x}{12+x}-4x=48$$

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Whakareatia -4x ki te \frac{12+x}{12+x}.

$$\frac{196x}{12+x}+\frac{-4x\left(12+x\right)}{12+x}=48$$

Tā te mea he rite te tauraro o \frac{196x}{12+x} me \frac{-4x\left(12+x\right)}{12+x}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$\frac{196x-4x\left(12+x\right)}{12+x}=48$$

Mahia ngā whakarea i roto o 196x-4x\left(12+x\right).

$$\frac{196x-48x-4x^{2}}{12+x}=48$$

Whakakotahitia ngā kupu rite i 196x-48x-4x^{2}.

$$\frac{148x-4x^{2}}{12+x}=48$$

Tē taea kia ōrite te tāupe x ki -12 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te x+12.

$$148x-4x^{2}=48\left(x+12\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 48 ki te x+12.

$$148x-4x^{2}=48x+576$$

Tangohia te 48x mai i ngā taha e rua.

$$148x-4x^{2}-48x=576$$

Pahekotia te 148x me -48x, ka 100x.

$$100x-4x^{2}=576$$

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.

$$-4x^{2}+100x=576$$

Whakawehea ngā taha e rua ki te -4.

$$\frac{-4x^{2}+100x}{-4}=\frac{576}{-4}$$

Mā te whakawehe ki te -4 ka wetekia te whakareanga ki te -4.

$$x^{2}+\frac{100}{-4}x=\frac{576}{-4}$$

Whakawehe 100 ki te -4.

$$x^{2}-25x=\frac{576}{-4}$$

Whakawehe 576 ki te -4.

$$x^{2}-25x=-144$$

Whakawehea te -25, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{25}{2}. Nā, tāpiria te pūrua o te -\frac{25}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-25x+\left(-\frac{25}{2}\right)^{2}=-144+\left(-\frac{25}{2}\right)^{2}$$

Pūruatia -\frac{25}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-25x+\frac{625}{4}=-144+\frac{625}{4}$$

Tāpiri -144 ki te \frac{625}{4}.

$$x^{2}-25x+\frac{625}{4}=\frac{49}{4}$$

Tauwehea x^{2}-25x+\frac{625}{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}.

$$\left(x-\frac{25}{2}\right)^{2}=\frac{49}{4}$$

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

$$\sqrt{\left(x-\frac{25}{2}\right)^{2}}=\sqrt{\frac{49}{4}}$$

Whakarūnātia.

$$x-\frac{25}{2}=\frac{7}{2} x-\frac{25}{2}=-\frac{7}{2}$$

Me tāpiri \frac{25}{2} ki ngā taha e rua o te whārite.

$$x=16 x=9$$