Whakaoti i te 1/4-y=1/4+1/y+2

$$\frac { 1 } { 4 - y } = \frac { 1 } { 4 } + \frac { 1 } { y + 2 }$$

Whakaoti mō y

$$y=-8$$

$$y=2$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe y ki tētahi o ngā uara -2,4 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 4\left(y-4\right)\left(y+2\right), arā, te tauraro pātahi he tino iti rawa te kitea o 4-y,4,y+2.

$$-8-4y=4\left(y-4\right)\left(y+2\right)\times \frac{1}{4}+4y-16$$

Whakareatia te 4 ki te \frac{1}{4}, ka 1.

$$-8-4y=\left(y-4\right)\left(y+2\right)+4y-16$$

Whakamahia te āhuatanga tuaritanga hei whakarea te y-4 ki te y+2 ka whakakotahi i ngā kupu rite.

$$-8-4y=y^{2}-2y-8+4y-16$$

Pahekotia te -2y me 4y, ka 2y.

$$-8-4y=y^{2}+2y-8-16$$

Tangohia te 16 i te -8, ka -24.

$$-8-4y=y^{2}+2y-24$$

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

$$-8-4y-y^{2}=2y-24$$

Tangohia te 2y mai i ngā taha e rua.

$$-8-4y-y^{2}-2y=-24$$

Pahekotia te -4y me -2y, ka -6y.

$$-8-6y-y^{2}=-24$$

Me tāpiri te 24 ki ngā taha e rua.

$$-8-6y-y^{2}+24=0$$

Tāpirihia te -8 ki te 24, ka 16.

$$16-6y-y^{2}=0$$

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.

$$-y^{2}-6y+16=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, -6 mō b, me 16 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-1\right)\times 16}}{2\left(-1\right)}$$

Pūrua -6.

$$y=\frac{-\left(-6\right)±\sqrt{36-4\left(-1\right)\times 16}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$y=\frac{-\left(-6\right)±\sqrt{36+4\times 16}}{2\left(-1\right)}$$

Whakareatia 4 ki te 16.

$$y=\frac{-\left(-6\right)±\sqrt{36+64}}{2\left(-1\right)}$$

Tāpiri 36 ki te 64.

$$y=\frac{-\left(-6\right)±\sqrt{100}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 100.

$$y=\frac{-\left(-6\right)±10}{2\left(-1\right)}$$

Ko te tauaro o -6 ko 6.

$$y=\frac{6±10}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$y=\frac{6±10}{-2}$$

Nā, me whakaoti te whārite y=\frac{6±10}{-2} ina he tāpiri te ±. Tāpiri 6 ki te 10.

$$y=\frac{16}{-2}$$

Whakawehe 16 ki te -2.

$$y=-8$$

Nā, me whakaoti te whārite y=\frac{6±10}{-2} ina he tango te ±. Tango 10 mai i 6.

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

Whakawehe -4 ki te -2.

$$y=2$$

Kua oti te whārite te whakatau.

$$y=-8 y=2$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe y ki tētahi o ngā uara -2,4 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 4\left(y-4\right)\left(y+2\right), arā, te tauraro pātahi he tino iti rawa te kitea o 4-y,4,y+2.

$$-8-4y=4\left(y-4\right)\left(y+2\right)\times \frac{1}{4}+4y-16$$

Whakareatia te 4 ki te \frac{1}{4}, ka 1.

$$-8-4y=\left(y-4\right)\left(y+2\right)+4y-16$$

Whakamahia te āhuatanga tuaritanga hei whakarea te y-4 ki te y+2 ka whakakotahi i ngā kupu rite.

$$-8-4y=y^{2}-2y-8+4y-16$$

Pahekotia te -2y me 4y, ka 2y.

$$-8-4y=y^{2}+2y-8-16$$

Tangohia te 16 i te -8, ka -24.

$$-8-4y=y^{2}+2y-24$$

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

$$-8-4y-y^{2}=2y-24$$

Tangohia te 2y mai i ngā taha e rua.

$$-8-4y-y^{2}-2y=-24$$

Pahekotia te -4y me -2y, ka -6y.

$$-8-6y-y^{2}=-24$$

Me tāpiri te 8 ki ngā taha e rua.

$$-6y-y^{2}=-24+8$$

Tāpirihia te -24 ki te 8, ka -16.

$$-6y-y^{2}=-16$$

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.

$$-y^{2}-6y=-16$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-y^{2}-6y}{-1}=-\frac{16}{-1}$$

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

$$y^{2}+\left(-\frac{6}{-1}\right)y=-\frac{16}{-1}$$

Whakawehe -6 ki te -1.

$$y^{2}+6y=-\frac{16}{-1}$$

Whakawehe -16 ki te -1.

$$y^{2}+6y=16$$

Whakawehea te 6, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 3. Nā, tāpiria te pūrua o te 3 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$y^{2}+6y+3^{2}=16+3^{2}$$

Pūrua 3.

$$y^{2}+6y+9=16+9$$

Tāpiri 16 ki te 9.

$$y^{2}+6y+9=25$$

Tauwehea y^{2}+6y+9. 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}.

$$\left(y+3\right)^{2}=25$$

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

$$\sqrt{\left(y+3\right)^{2}}=\sqrt{25}$$

Whakarūnātia.

$$y+3=5 y+3=-5$$

Me tango 3 mai i ngā taha e rua o te whārite.

$$y=2 y=-8$$