Whakaoti i te 11y^2+y=2

$$11 y ^ { 2 } + y = 2$$

Whakaoti mō y

$$y=\frac{\sqrt{89}-1}{22}\approx 0.383362779$$

$$y=\frac{-\sqrt{89}-1}{22}\approx -0.47427187$$

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.

$$11y^{2}+y=2$$

Me tango 2 mai i ngā taha e rua o te whārite.

$$11y^{2}+y-2=2-2$$

Mā te tango i te 2 i a ia ake anō ka toe ko te 0.

$$11y^{2}+y-2=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 11 mō a, 1 mō b, me -2 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-1±\sqrt{1^{2}-4\times 11\left(-2\right)}}{2\times 11}$$

Pūrua 1.

$$y=\frac{-1±\sqrt{1-4\times 11\left(-2\right)}}{2\times 11}$$

Whakareatia -4 ki te 11.

$$y=\frac{-1±\sqrt{1-44\left(-2\right)}}{2\times 11}$$

Whakareatia -44 ki te -2.

$$y=\frac{-1±\sqrt{1+88}}{2\times 11}$$

Tāpiri 1 ki te 88.

$$y=\frac{-1±\sqrt{89}}{2\times 11}$$

Whakareatia 2 ki te 11.

$$y=\frac{-1±\sqrt{89}}{22}$$

Nā, me whakaoti te whārite y=\frac{-1±\sqrt{89}}{22} ina he tāpiri te ±. Tāpiri -1 ki te \sqrt{89}.

$$y=\frac{\sqrt{89}-1}{22}$$

Nā, me whakaoti te whārite y=\frac{-1±\sqrt{89}}{22} ina he tango te ±. Tango \sqrt{89} mai i -1.

$$y=\frac{-\sqrt{89}-1}{22}$$

Kua oti te whārite te whakatau.

$$y=\frac{\sqrt{89}-1}{22} y=\frac{-\sqrt{89}-1}{22}$$

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.

$$11y^{2}+y=2$$

Whakawehea ngā taha e rua ki te 11.

$$\frac{11y^{2}+y}{11}=\frac{2}{11}$$

Mā te whakawehe ki te 11 ka wetekia te whakareanga ki te 11.

$$y^{2}+\frac{1}{11}y=\frac{2}{11}$$

Whakawehea te \frac{1}{11}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{1}{22}. Nā, tāpiria te pūrua o te \frac{1}{22} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$y^{2}+\frac{1}{11}y+\left(\frac{1}{22}\right)^{2}=\frac{2}{11}+\left(\frac{1}{22}\right)^{2}$$

Pūruatia \frac{1}{22} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$y^{2}+\frac{1}{11}y+\frac{1}{484}=\frac{2}{11}+\frac{1}{484}$$

Tāpiri \frac{2}{11} ki te \frac{1}{484} 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.

$$y^{2}+\frac{1}{11}y+\frac{1}{484}=\frac{89}{484}$$

Tauwehea y^{2}+\frac{1}{11}y+\frac{1}{484}. 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}.

$$\left(y+\frac{1}{22}\right)^{2}=\frac{89}{484}$$

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

$$\sqrt{\left(y+\frac{1}{22}\right)^{2}}=\sqrt{\frac{89}{484}}$$

Whakarūnātia.

$$y+\frac{1}{22}=\frac{\sqrt{89}}{22} y+\frac{1}{22}=-\frac{\sqrt{89}}{22}$$

Me tango \frac{1}{22} mai i ngā taha e rua o te whārite.

$$y=\frac{\sqrt{89}-1}{22} y=\frac{-\sqrt{89}-1}{22}$$