Whakaoti i te -2y^2-6y+5=0

$$- 2 y ^ { 2 } - 6 y + 5 = 0$$

Whakaoti mō y

$$y=\frac{\sqrt{19}-3}{2}\approx 0.679449472$$

$$y=\frac{-\sqrt{19}-3}{2}\approx -3.679449472$$

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.

$$-2y^{2}-6y+5=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -2 mō a, -6 mō b, me 5 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-2\right)\times 5}}{2\left(-2\right)}$$

Pūrua -6.

$$y=\frac{-\left(-6\right)±\sqrt{36-4\left(-2\right)\times 5}}{2\left(-2\right)}$$

Whakareatia -4 ki te -2.

$$y=\frac{-\left(-6\right)±\sqrt{36+8\times 5}}{2\left(-2\right)}$$

Whakareatia 8 ki te 5.

$$y=\frac{-\left(-6\right)±\sqrt{36+40}}{2\left(-2\right)}$$

Tāpiri 36 ki te 40.

$$y=\frac{-\left(-6\right)±\sqrt{76}}{2\left(-2\right)}$$

Tuhia te pūtakerua o te 76.

$$y=\frac{-\left(-6\right)±2\sqrt{19}}{2\left(-2\right)}$$

Ko te tauaro o -6 ko 6.

$$y=\frac{6±2\sqrt{19}}{2\left(-2\right)}$$

Whakareatia 2 ki te -2.

$$y=\frac{6±2\sqrt{19}}{-4}$$

Nā, me whakaoti te whārite y=\frac{6±2\sqrt{19}}{-4} ina he tāpiri te ±. Tāpiri 6 ki te 2\sqrt{19}.

$$y=\frac{2\sqrt{19}+6}{-4}$$

Whakawehe 6+2\sqrt{19} ki te -4.

$$y=\frac{-\sqrt{19}-3}{2}$$

Nā, me whakaoti te whārite y=\frac{6±2\sqrt{19}}{-4} ina he tango te ±. Tango 2\sqrt{19} mai i 6.

$$y=\frac{6-2\sqrt{19}}{-4}$$

Whakawehe 6-2\sqrt{19} ki te -4.

$$y=\frac{\sqrt{19}-3}{2}$$

Kua oti te whārite te whakatau.

$$y=\frac{-\sqrt{19}-3}{2} y=\frac{\sqrt{19}-3}{2}$$

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.

$$-2y^{2}-6y+5=0$$

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

$$-2y^{2}-6y+5-5=-5$$

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

$$-2y^{2}-6y=-5$$

Whakawehea ngā taha e rua ki te -2.

$$\frac{-2y^{2}-6y}{-2}=-\frac{5}{-2}$$

Mā te whakawehe ki te -2 ka wetekia te whakareanga ki te -2.

$$y^{2}+\left(-\frac{6}{-2}\right)y=-\frac{5}{-2}$$

Whakawehe -6 ki te -2.

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

Whakawehe -5 ki te -2.

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

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.

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

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

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

Tāpiri \frac{5}{2} ki te \frac{9}{4} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$y^{2}+3y+\frac{9}{4}=\frac{19}{4}$$

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

$$\left(y+\frac{3}{2}\right)^{2}=\frac{19}{4}$$

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

$$\sqrt{\left(y+\frac{3}{2}\right)^{2}}=\sqrt{\frac{19}{4}}$$

Whakarūnātia.

$$y+\frac{3}{2}=\frac{\sqrt{19}}{2} y+\frac{3}{2}=-\frac{\sqrt{19}}{2}$$

Me tango \frac{3}{2} mai i ngā taha e rua o te whārite.

$$y=\frac{\sqrt{19}-3}{2} y=\frac{-\sqrt{19}-3}{2}$$