Whakaoti i te (-y^2+3y+5=0)

$$(- { y }^{ 2 } +3y+5=0)$$

Whakaoti mō y

$$y = \frac{\sqrt{29} + 3}{2} \approx 4.192582404$$

$$y=\frac{3-\sqrt{29}}{2}\approx -1.192582404$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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.

$$-y^{2}+3y+5=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 3 mō b, me 5 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-3±\sqrt{3^{2}-4\left(-1\right)\times 5}}{2\left(-1\right)}$$

Pūrua 3.

$$y=\frac{-3±\sqrt{9-4\left(-1\right)\times 5}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$y=\frac{-3±\sqrt{9+4\times 5}}{2\left(-1\right)}$$

Whakareatia 4 ki te 5.

$$y=\frac{-3±\sqrt{9+20}}{2\left(-1\right)}$$

Tāpiri 9 ki te 20.

$$y=\frac{-3±\sqrt{29}}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$y=\frac{-3±\sqrt{29}}{-2}$$

Nā, me whakaoti te whārite y=\frac{-3±\sqrt{29}}{-2} ina he tāpiri te ±. Tāpiri -3 ki te \sqrt{29}.

$$y=\frac{\sqrt{29}-3}{-2}$$

Whakawehe -3+\sqrt{29} ki te -2.

$$y=\frac{3-\sqrt{29}}{2}$$

Nā, me whakaoti te whārite y=\frac{-3±\sqrt{29}}{-2} ina he tango te ±. Tango \sqrt{29} mai i -3.

$$y=\frac{-\sqrt{29}-3}{-2}$$

Whakawehe -3-\sqrt{29} ki te -2.

$$y=\frac{\sqrt{29}+3}{2}$$

Kua oti te whārite te whakatau.

$$y=\frac{3-\sqrt{29}}{2} y=\frac{\sqrt{29}+3}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

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.

$$-y^{2}+3y+5=0$$

Me tango 5 mai i ngā taha e rua o te whārite.

$$-y^{2}+3y+5-5=-5$$

Mā te tango i te 5 i a ia ake anō ka toe ko te 0.

$$-y^{2}+3y=-5$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-y^{2}+3y}{-1}=-\frac{5}{-1}$$

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

$$y^{2}+\frac{3}{-1}y=-\frac{5}{-1}$$

Whakawehe 3 ki te -1.

$$y^{2}-3y=-\frac{5}{-1}$$

Whakawehe -5 ki te -1.

$$y^{2}-3y=5$$

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

$$y^{2}-3y+\left(-\frac{3}{2}\right)^{2}=5+\left(-\frac{3}{2}\right)^{2}$$

Pūruatia -\frac{3}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$y^{2}-3y+\frac{9}{4}=5+\frac{9}{4}$$

Tāpiri 5 ki te \frac{9}{4}.

$$y^{2}-3y+\frac{9}{4}=\frac{29}{4}$$

Tauwehea te y^{2}-3y+\frac{9}{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}.

$$\left(y-\frac{3}{2}\right)^{2}=\frac{29}{4}$$

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

$$\sqrt{\left(y-\frac{3}{2}\right)^{2}}=\sqrt{\frac{29}{4}}$$

Whakarūnātia.

$$y-\frac{3}{2}=\frac{\sqrt{29}}{2} y-\frac{3}{2}=-\frac{\sqrt{29}}{2}$$

Me tāpiri \frac{3}{2} ki ngā taha e rua o te whārite.

$$y=\frac{\sqrt{29}+3}{2} y=\frac{3-\sqrt{29}}{2}$$