Whakaoti i te u+2/u-4-1=u+1/u-3

$$\frac { u + 2 } { u - 4 } - 1 = \frac { u + 1 } { u - 3 }$$

Whakaoti mō u

$$u=2$$

$$u=7$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe u ki tētahi o ngā uara 3,4 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(u-4\right)\left(u-3\right), arā, te tauraro pātahi he tino iti rawa te kitea o u-4,u-3.

$$\left(u-3\right)\left(u+2\right)+\left(u-4\right)\left(u-3\right)\left(-1\right)=\left(u-4\right)\left(u+1\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te u-3 ki te u+2 ka whakakotahi i ngā kupu rite.

$$u^{2}-u-6+\left(u-4\right)\left(u-3\right)\left(-1\right)=\left(u-4\right)\left(u+1\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te u-4 ki te u-3 ka whakakotahi i ngā kupu rite.

$$u^{2}-u-6+\left(u^{2}-7u+12\right)\left(-1\right)=\left(u-4\right)\left(u+1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te u^{2}-7u+12 ki te -1.

$$u^{2}-u-6-u^{2}+7u-12=\left(u-4\right)\left(u+1\right)$$

Pahekotia te u^{2} me -u^{2}, ka 0.

$$-u-6+7u-12=\left(u-4\right)\left(u+1\right)$$

Pahekotia te -u me 7u, ka 6u.

$$6u-6-12=\left(u-4\right)\left(u+1\right)$$

Tangohia te 12 i te -6, ka -18.

$$6u-18=\left(u-4\right)\left(u+1\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te u-4 ki te u+1 ka whakakotahi i ngā kupu rite.

$$6u-18=u^{2}-3u-4$$

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

$$6u-18-u^{2}=-3u-4$$

Me tāpiri te 3u ki ngā taha e rua.

$$6u-18-u^{2}+3u=-4$$

Pahekotia te 6u me 3u, ka 9u.

$$9u-18-u^{2}=-4$$

Me tāpiri te 4 ki ngā taha e rua.

$$9u-18-u^{2}+4=0$$

Tāpirihia te -18 ki te 4, ka -14.

$$9u-14-u^{2}=0$$

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.

$$-u^{2}+9u-14=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 9 mō b, me -14 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$u=\frac{-9±\sqrt{9^{2}-4\left(-1\right)\left(-14\right)}}{2\left(-1\right)}$$

Pūrua 9.

$$u=\frac{-9±\sqrt{81-4\left(-1\right)\left(-14\right)}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$u=\frac{-9±\sqrt{81+4\left(-14\right)}}{2\left(-1\right)}$$

Whakareatia 4 ki te -14.

$$u=\frac{-9±\sqrt{81-56}}{2\left(-1\right)}$$

Tāpiri 81 ki te -56.

$$u=\frac{-9±\sqrt{25}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 25.

$$u=\frac{-9±5}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$u=\frac{-9±5}{-2}$$

Nā, me whakaoti te whārite u=\frac{-9±5}{-2} ina he tāpiri te ±. Tāpiri -9 ki te 5.

$$u=-\frac{4}{-2}$$

Whakawehe -4 ki te -2.

$$u=2$$

Nā, me whakaoti te whārite u=\frac{-9±5}{-2} ina he tango te ±. Tango 5 mai i -9.

$$u=-\frac{14}{-2}$$

Whakawehe -14 ki te -2.

$$u=7$$

Kua oti te whārite te whakatau.

$$u=2 u=7$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe u ki tētahi o ngā uara 3,4 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(u-4\right)\left(u-3\right), arā, te tauraro pātahi he tino iti rawa te kitea o u-4,u-3.

$$\left(u-3\right)\left(u+2\right)+\left(u-4\right)\left(u-3\right)\left(-1\right)=\left(u-4\right)\left(u+1\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te u-3 ki te u+2 ka whakakotahi i ngā kupu rite.

$$u^{2}-u-6+\left(u-4\right)\left(u-3\right)\left(-1\right)=\left(u-4\right)\left(u+1\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te u-4 ki te u-3 ka whakakotahi i ngā kupu rite.

$$u^{2}-u-6+\left(u^{2}-7u+12\right)\left(-1\right)=\left(u-4\right)\left(u+1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te u^{2}-7u+12 ki te -1.

$$u^{2}-u-6-u^{2}+7u-12=\left(u-4\right)\left(u+1\right)$$

Pahekotia te u^{2} me -u^{2}, ka 0.

$$-u-6+7u-12=\left(u-4\right)\left(u+1\right)$$

Pahekotia te -u me 7u, ka 6u.

$$6u-6-12=\left(u-4\right)\left(u+1\right)$$

Tangohia te 12 i te -6, ka -18.

$$6u-18=\left(u-4\right)\left(u+1\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te u-4 ki te u+1 ka whakakotahi i ngā kupu rite.

$$6u-18=u^{2}-3u-4$$

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

$$6u-18-u^{2}=-3u-4$$

Me tāpiri te 3u ki ngā taha e rua.

$$6u-18-u^{2}+3u=-4$$

Pahekotia te 6u me 3u, ka 9u.

$$9u-18-u^{2}=-4$$

Me tāpiri te 18 ki ngā taha e rua.

$$9u-u^{2}=-4+18$$

Tāpirihia te -4 ki te 18, ka 14.

$$9u-u^{2}=14$$

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.

$$-u^{2}+9u=14$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-u^{2}+9u}{-1}=\frac{14}{-1}$$

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

$$u^{2}+\frac{9}{-1}u=\frac{14}{-1}$$

Whakawehe 9 ki te -1.

$$u^{2}-9u=\frac{14}{-1}$$

Whakawehe 14 ki te -1.

$$u^{2}-9u=-14$$

Whakawehea te -9, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{9}{2}. Nā, tāpiria te pūrua o te -\frac{9}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$u^{2}-9u+\left(-\frac{9}{2}\right)^{2}=-14+\left(-\frac{9}{2}\right)^{2}$$

Pūruatia -\frac{9}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$u^{2}-9u+\frac{81}{4}=-14+\frac{81}{4}$$

Tāpiri -14 ki te \frac{81}{4}.

$$u^{2}-9u+\frac{81}{4}=\frac{25}{4}$$

Tauwehea te u^{2}-9u+\frac{81}{4}. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(u-\frac{9}{2}\right)^{2}=\frac{25}{4}$$

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

$$\sqrt{\left(u-\frac{9}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Whakarūnātia.

$$u-\frac{9}{2}=\frac{5}{2} u-\frac{9}{2}=-\frac{5}{2}$$

Me tāpiri \frac{9}{2} ki ngā taha e rua o te whārite.

$$u=7 u=2$$