Whakaoti i te 4y^2-13y+36=0

$$4 y ^ { 2 } - 13 y + 36 = 0$$

Whakaoti mō y

$$y=\frac{13+\sqrt{407}i}{8}\approx 1.625+2.521780125i$$

$$y=\frac{-\sqrt{407}i+13}{8}\approx 1.625-2.521780125i$$

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.

$$4y^{2}-13y+36=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, -13 mō b, me 36 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\times 4\times 36}}{2\times 4}$$

Pūrua -13.

$$y=\frac{-\left(-13\right)±\sqrt{169-4\times 4\times 36}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$y=\frac{-\left(-13\right)±\sqrt{169-16\times 36}}{2\times 4}$$

Whakareatia -16 ki te 36.

$$y=\frac{-\left(-13\right)±\sqrt{169-576}}{2\times 4}$$

Tāpiri 169 ki te -576.

$$y=\frac{-\left(-13\right)±\sqrt{-407}}{2\times 4}$$

Tuhia te pūtakerua o te -407.

$$y=\frac{-\left(-13\right)±\sqrt{407}i}{2\times 4}$$

Ko te tauaro o -13 ko 13.

$$y=\frac{13±\sqrt{407}i}{2\times 4}$$

Whakareatia 2 ki te 4.

$$y=\frac{13±\sqrt{407}i}{8}$$

Nā, me whakaoti te whārite y=\frac{13±\sqrt{407}i}{8} ina he tāpiri te ±. Tāpiri 13 ki te i\sqrt{407}.

$$y=\frac{13+\sqrt{407}i}{8}$$

Nā, me whakaoti te whārite y=\frac{13±\sqrt{407}i}{8} ina he tango te ±. Tango i\sqrt{407} mai i 13.

$$y=\frac{-\sqrt{407}i+13}{8}$$

Kua oti te whārite te whakatau.

$$y=\frac{13+\sqrt{407}i}{8} y=\frac{-\sqrt{407}i+13}{8}$$

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.

$$4y^{2}-13y+36=0$$

Me tango 36 mai i ngā taha e rua o te whārite.

$$4y^{2}-13y+36-36=-36$$

Mā te tango i te 36 i a ia ake anō ka toe ko te 0.

$$4y^{2}-13y=-36$$

Whakawehea ngā taha e rua ki te 4.

$$\frac{4y^{2}-13y}{4}=-\frac{36}{4}$$

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

$$y^{2}-\frac{13}{4}y=-\frac{36}{4}$$

Whakawehe -36 ki te 4.

$$y^{2}-\frac{13}{4}y=-9$$

Whakawehea te -\frac{13}{4}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{13}{8}. Nā, tāpiria te pūrua o te -\frac{13}{8} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$y^{2}-\frac{13}{4}y+\left(-\frac{13}{8}\right)^{2}=-9+\left(-\frac{13}{8}\right)^{2}$$

Pūruatia -\frac{13}{8} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$y^{2}-\frac{13}{4}y+\frac{169}{64}=-9+\frac{169}{64}$$

Tāpiri -9 ki te \frac{169}{64}.

$$y^{2}-\frac{13}{4}y+\frac{169}{64}=-\frac{407}{64}$$

Tauwehea y^{2}-\frac{13}{4}y+\frac{169}{64}. 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}.

$$\left(y-\frac{13}{8}\right)^{2}=-\frac{407}{64}$$

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

$$\sqrt{\left(y-\frac{13}{8}\right)^{2}}=\sqrt{-\frac{407}{64}}$$

Whakarūnātia.

$$y-\frac{13}{8}=\frac{\sqrt{407}i}{8} y-\frac{13}{8}=-\frac{\sqrt{407}i}{8}$$

Me tāpiri \frac{13}{8} ki ngā taha e rua o te whārite.

$$y=\frac{13+\sqrt{407}i}{8} y=\frac{-\sqrt{407}i+13}{8}$$