Whakaoti i te 5y^2-90y+54=0

$$5 { y }^{ 2 } -90y+54 = 0$$

Whakaoti mō y

$$y=\frac{3\sqrt{195}}{5}+9\approx 17.378544026$$

$$y=-\frac{3\sqrt{195}}{5}+9\approx 0.621455974$$

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.

$$5y^{2}-90y+54=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 5 mō a, -90 mō b, me 54 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-90\right)±\sqrt{\left(-90\right)^{2}-4\times 5\times 54}}{2\times 5}$$

Pūrua -90.

$$y=\frac{-\left(-90\right)±\sqrt{8100-4\times 5\times 54}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$y=\frac{-\left(-90\right)±\sqrt{8100-20\times 54}}{2\times 5}$$

Whakareatia -20 ki te 54.

$$y=\frac{-\left(-90\right)±\sqrt{8100-1080}}{2\times 5}$$

Tāpiri 8100 ki te -1080.

$$y=\frac{-\left(-90\right)±\sqrt{7020}}{2\times 5}$$

Tuhia te pūtakerua o te 7020.

$$y=\frac{-\left(-90\right)±6\sqrt{195}}{2\times 5}$$

Ko te tauaro o -90 ko 90.

$$y=\frac{90±6\sqrt{195}}{2\times 5}$$

Whakareatia 2 ki te 5.

$$y=\frac{90±6\sqrt{195}}{10}$$

Nā, me whakaoti te whārite y=\frac{90±6\sqrt{195}}{10} ina he tāpiri te ±. Tāpiri 90 ki te 6\sqrt{195}.

$$y=\frac{6\sqrt{195}+90}{10}$$

Whakawehe 90+6\sqrt{195} ki te 10.

$$y=\frac{3\sqrt{195}}{5}+9$$

Nā, me whakaoti te whārite y=\frac{90±6\sqrt{195}}{10} ina he tango te ±. Tango 6\sqrt{195} mai i 90.

$$y=\frac{90-6\sqrt{195}}{10}$$

Whakawehe 90-6\sqrt{195} ki te 10.

$$y=-\frac{3\sqrt{195}}{5}+9$$

Kua oti te whārite te whakatau.

$$y=\frac{3\sqrt{195}}{5}+9 y=-\frac{3\sqrt{195}}{5}+9$$

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.

$$5y^{2}-90y+54=0$$

Me tango 54 mai i ngā taha e rua o te whārite.

$$5y^{2}-90y+54-54=-54$$

Mā te tango i te 54 i a ia ake anō ka toe ko te 0.

$$5y^{2}-90y=-54$$

Whakawehea ngā taha e rua ki te 5.

$$\frac{5y^{2}-90y}{5}=-\frac{54}{5}$$

Mā te whakawehe ki te 5 ka wetekia te whakareanga ki te 5.

$$y^{2}+\left(-\frac{90}{5}\right)y=-\frac{54}{5}$$

Whakawehe -90 ki te 5.

$$y^{2}-18y=-\frac{54}{5}$$

Whakawehea te -18, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -9. Nā, tāpiria te pūrua o te -9 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$y^{2}-18y+\left(-9\right)^{2}=-\frac{54}{5}+\left(-9\right)^{2}$$

Pūrua -9.

$$y^{2}-18y+81=-\frac{54}{5}+81$$

Tāpiri -\frac{54}{5} ki te 81.

$$y^{2}-18y+81=\frac{351}{5}$$

Tauwehea y^{2}-18y+81. 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}.

$$\left(y-9\right)^{2}=\frac{351}{5}$$

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

$$\sqrt{\left(y-9\right)^{2}}=\sqrt{\frac{351}{5}}$$

Whakarūnātia.

$$y-9=\frac{3\sqrt{195}}{5} y-9=-\frac{3\sqrt{195}}{5}$$

Me tāpiri 9 ki ngā taha e rua o te whārite.

$$y=\frac{3\sqrt{195}}{5}+9 y=-\frac{3\sqrt{195}}{5}+9$$