Whakaoti i te 6y^2+13y+63=0

$$6 y ^ { 2 } + 13 y + 63 = 0$$

Whakaoti mō y

$$y=\frac{-13+\sqrt{1343}i}{12}\approx -1.083333333+3.0539137i$$

$$y=\frac{-\sqrt{1343}i-13}{12}\approx -1.083333333-3.0539137i$$

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.

$$6y^{2}+13y+63=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 6 mō a, 13 mō b, me 63 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-13±\sqrt{13^{2}-4\times 6\times 63}}{2\times 6}$$

Pūrua 13.

$$y=\frac{-13±\sqrt{169-4\times 6\times 63}}{2\times 6}$$

Whakareatia -4 ki te 6.

$$y=\frac{-13±\sqrt{169-24\times 63}}{2\times 6}$$

Whakareatia -24 ki te 63.

$$y=\frac{-13±\sqrt{169-1512}}{2\times 6}$$

Tāpiri 169 ki te -1512.

$$y=\frac{-13±\sqrt{-1343}}{2\times 6}$$

Tuhia te pūtakerua o te -1343.

$$y=\frac{-13±\sqrt{1343}i}{2\times 6}$$

Whakareatia 2 ki te 6.

$$y=\frac{-13±\sqrt{1343}i}{12}$$

Nā, me whakaoti te whārite y=\frac{-13±\sqrt{1343}i}{12} ina he tāpiri te ±. Tāpiri -13 ki te i\sqrt{1343}.

$$y=\frac{-13+\sqrt{1343}i}{12}$$

Nā, me whakaoti te whārite y=\frac{-13±\sqrt{1343}i}{12} ina he tango te ±. Tango i\sqrt{1343} mai i -13.

$$y=\frac{-\sqrt{1343}i-13}{12}$$

Kua oti te whārite te whakatau.

$$y=\frac{-13+\sqrt{1343}i}{12} y=\frac{-\sqrt{1343}i-13}{12}$$

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.

$$6y^{2}+13y+63=0$$

Me tango 63 mai i ngā taha e rua o te whārite.

$$6y^{2}+13y+63-63=-63$$

Mā te tango i te 63 i a ia ake anō ka toe ko te 0.

$$6y^{2}+13y=-63$$

Whakawehea ngā taha e rua ki te 6.

$$\frac{6y^{2}+13y}{6}=-\frac{63}{6}$$

Mā te whakawehe ki te 6 ka wetekia te whakareanga ki te 6.

$$y^{2}+\frac{13}{6}y=-\frac{63}{6}$$

Whakahekea te hautanga \frac{-63}{6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 3.

$$y^{2}+\frac{13}{6}y=-\frac{21}{2}$$

Whakawehea te \frac{13}{6}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{13}{12}. Nā, tāpiria te pūrua o te \frac{13}{12} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$y^{2}+\frac{13}{6}y+\left(\frac{13}{12}\right)^{2}=-\frac{21}{2}+\left(\frac{13}{12}\right)^{2}$$

Pūruatia \frac{13}{12} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$y^{2}+\frac{13}{6}y+\frac{169}{144}=-\frac{21}{2}+\frac{169}{144}$$

Tāpiri -\frac{21}{2} ki te \frac{169}{144} 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.

$$y^{2}+\frac{13}{6}y+\frac{169}{144}=-\frac{1343}{144}$$

Tauwehea y^{2}+\frac{13}{6}y+\frac{169}{144}. 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}.

$$\left(y+\frac{13}{12}\right)^{2}=-\frac{1343}{144}$$

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

$$\sqrt{\left(y+\frac{13}{12}\right)^{2}}=\sqrt{-\frac{1343}{144}}$$

Whakarūnātia.

$$y+\frac{13}{12}=\frac{\sqrt{1343}i}{12} y+\frac{13}{12}=-\frac{\sqrt{1343}i}{12}$$

Me tango \frac{13}{12} mai i ngā taha e rua o te whārite.

$$y=\frac{-13+\sqrt{1343}i}{12} y=\frac{-\sqrt{1343}i-13}{12}$$