Whakaoti i te y^2-3y=9

$$y ^ { 2 } - 3 y = 9$$

Whakaoti mō y

$$y = \frac{3 \sqrt{5} + 3}{2} \approx 4.854101966$$

$$y=\frac{3-3\sqrt{5}}{2}\approx -1.854101966$$

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.

$$y^{2}-3y=9$$

Me tango 9 mai i ngā taha e rua o te whārite.

$$y^{2}-3y-9=9-9$$

Mā te tango i te 9 i a ia ake anō ka toe ko te 0.

$$y^{2}-3y-9=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -3 mō b, me -9 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Pūrua -3.

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

Whakareatia -4 ki te -9.

$$y=\frac{-\left(-3\right)±\sqrt{9+36}}{2}$$

Tāpiri 9 ki te 36.

$$y=\frac{-\left(-3\right)±\sqrt{45}}{2}$$

Tuhia te pūtakerua o te 45.

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

Ko te tauaro o -3 ko 3.

$$y=\frac{3±3\sqrt{5}}{2}$$

Nā, me whakaoti te whārite y=\frac{3±3\sqrt{5}}{2} ina he tāpiri te ±. Tāpiri 3 ki te 3\sqrt{5}.

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

Nā, me whakaoti te whārite y=\frac{3±3\sqrt{5}}{2} ina he tango te ±. Tango 3\sqrt{5} mai i 3.

$$y=\frac{3-3\sqrt{5}}{2}$$

Kua oti te whārite te whakatau.

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

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.

$$y^{2}-3y=9$$

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.

$$y^{2}-3y+\left(-\frac{3}{2}\right)^{2}=9+\left(-\frac{3}{2}\right)^{2}$$

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

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

Tāpiri 9 ki te \frac{9}{4}.

$$y^{2}-3y+\frac{9}{4}=\frac{45}{4}$$

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}.

$$\left(y-\frac{3}{2}\right)^{2}=\frac{45}{4}$$

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

$$\sqrt{\left(y-\frac{3}{2}\right)^{2}}=\sqrt{\frac{45}{4}}$$

Whakarūnātia.

$$y-\frac{3}{2}=\frac{3\sqrt{5}}{2} y-\frac{3}{2}=-\frac{3\sqrt{5}}{2}$$

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

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