Whakaoti i te y=(y-4)(y+4)

$$y = ( y - 4 ) ( y + 4 )$$

Whakaoti mō y

$$y = \frac{\sqrt{65} + 1}{2} \approx 4.531128874$$

$$y=\frac{1-\sqrt{65}}{2}\approx -3.531128874$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakaarohia te \left(y-4\right)\left(y+4\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 4.

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

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

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

Me tāpiri te 16 ki ngā taha e rua.

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

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.

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

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

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

Pūrua 1.

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

Whakareatia -4 ki te -1.

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

Whakareatia 4 ki te 16.

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

Tāpiri 1 ki te 64.

$$y=\frac{-1±\sqrt{65}}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$y=\frac{-1±\sqrt{65}}{-2}$$

Nā, me whakaoti te whārite y=\frac{-1±\sqrt{65}}{-2} ina he tāpiri te ±. Tāpiri -1 ki te \sqrt{65}.

$$y=\frac{\sqrt{65}-1}{-2}$$

Whakawehe -1+\sqrt{65} ki te -2.

$$y=\frac{1-\sqrt{65}}{2}$$

Nā, me whakaoti te whārite y=\frac{-1±\sqrt{65}}{-2} ina he tango te ±. Tango \sqrt{65} mai i -1.

$$y=\frac{-\sqrt{65}-1}{-2}$$

Whakawehe -1-\sqrt{65} ki te -2.

$$y=\frac{\sqrt{65}+1}{2}$$

Kua oti te whārite te whakatau.

$$y=\frac{1-\sqrt{65}}{2} y=\frac{\sqrt{65}+1}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakaarohia te \left(y-4\right)\left(y+4\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 4.

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

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

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

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.

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

Whakawehea ngā taha e rua ki te -1.

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

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

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

Whakawehe 1 ki te -1.

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

Whakawehe -16 ki te -1.

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

Whakawehea te -1, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{2}. Nā, tāpiria te pūrua o te -\frac{1}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

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

Pūruatia -\frac{1}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$y^{2}-y+\frac{1}{4}=16+\frac{1}{4}$$

Tāpiri 16 ki te \frac{1}{4}.

$$y^{2}-y+\frac{1}{4}=\frac{65}{4}$$

Tauwehea y^{2}-y+\frac{1}{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}.

$$\left(y-\frac{1}{2}\right)^{2}=\frac{65}{4}$$

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

$$\sqrt{\left(y-\frac{1}{2}\right)^{2}}=\sqrt{\frac{65}{4}}$$

Whakarūnātia.

$$y-\frac{1}{2}=\frac{\sqrt{65}}{2} y-\frac{1}{2}=-\frac{\sqrt{65}}{2}$$

Me tāpiri \frac{1}{2} ki ngā taha e rua o te whārite.

$$y=\frac{\sqrt{65}+1}{2} y=\frac{1-\sqrt{65}}{2}$$