Whakaoti i te 4y^2+24y-374=0

$$4 y ^ { 2 } + 24 y - 374 = 0$$

Whakaoti mō y

$$y=\frac{\sqrt{410}}{2}-3\approx 7.124228366$$

$$y=-\frac{\sqrt{410}}{2}-3\approx -13.124228366$$

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.

$$4y^{2}+24y-374=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, 24 mō b, me -374 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-24±\sqrt{24^{2}-4\times 4\left(-374\right)}}{2\times 4}$$

Pūrua 24.

$$y=\frac{-24±\sqrt{576-4\times 4\left(-374\right)}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$y=\frac{-24±\sqrt{576-16\left(-374\right)}}{2\times 4}$$

Whakareatia -16 ki te -374.

$$y=\frac{-24±\sqrt{576+5984}}{2\times 4}$$

Tāpiri 576 ki te 5984.

$$y=\frac{-24±\sqrt{6560}}{2\times 4}$$

Tuhia te pūtakerua o te 6560.

$$y=\frac{-24±4\sqrt{410}}{2\times 4}$$

Whakareatia 2 ki te 4.

$$y=\frac{-24±4\sqrt{410}}{8}$$

Nā, me whakaoti te whārite y=\frac{-24±4\sqrt{410}}{8} ina he tāpiri te ±. Tāpiri -24 ki te 4\sqrt{410}.

$$y=\frac{4\sqrt{410}-24}{8}$$

Whakawehe -24+4\sqrt{410} ki te 8.

$$y=\frac{\sqrt{410}}{2}-3$$

Nā, me whakaoti te whārite y=\frac{-24±4\sqrt{410}}{8} ina he tango te ±. Tango 4\sqrt{410} mai i -24.

$$y=\frac{-4\sqrt{410}-24}{8}$$

Whakawehe -24-4\sqrt{410} ki te 8.

$$y=-\frac{\sqrt{410}}{2}-3$$

Kua oti te whārite te whakatau.

$$y=\frac{\sqrt{410}}{2}-3 y=-\frac{\sqrt{410}}{2}-3$$

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.

$$4y^{2}+24y-374=0$$

Me tāpiri 374 ki ngā taha e rua o te whārite.

$$4y^{2}+24y-374-\left(-374\right)=-\left(-374\right)$$

Mā te tango i te -374 i a ia ake anō ka toe ko te 0.

$$4y^{2}+24y=-\left(-374\right)$$

Tango -374 mai i 0.

$$4y^{2}+24y=374$$

Whakawehea ngā taha e rua ki te 4.

$$\frac{4y^{2}+24y}{4}=\frac{374}{4}$$

Mā te whakawehe ki te 4 ka wetekia te whakareanga ki te 4.

$$y^{2}+\frac{24}{4}y=\frac{374}{4}$$

Whakawehe 24 ki te 4.

$$y^{2}+6y=\frac{374}{4}$$

Whakahekea te hautanga \frac{374}{4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$y^{2}+6y=\frac{187}{2}$$

Whakawehea te 6, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 3. Nā, tāpiria te pūrua o te 3 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$y^{2}+6y+3^{2}=\frac{187}{2}+3^{2}$$

Pūrua 3.

$$y^{2}+6y+9=\frac{187}{2}+9$$

Tāpiri \frac{187}{2} ki te 9.

$$y^{2}+6y+9=\frac{205}{2}$$

Tauwehea y^{2}+6y+9. 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}.

$$\left(y+3\right)^{2}=\frac{205}{2}$$

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

$$\sqrt{\left(y+3\right)^{2}}=\sqrt{\frac{205}{2}}$$

Whakarūnātia.

$$y+3=\frac{\sqrt{410}}{2} y+3=-\frac{\sqrt{410}}{2}$$

Me tango 3 mai i ngā taha e rua o te whārite.

$$y=\frac{\sqrt{410}}{2}-3 y=-\frac{\sqrt{410}}{2}-3$$