Whakaoti i te 11x^2+4x-2=0

$$11 { x }^{ 2 } +4x-2=0$$

Whakaoti mō x

$$x=\frac{\sqrt{26}-2}{11}\approx 0.281729047$$

$$x=\frac{-\sqrt{26}-2}{11}\approx -0.64536541$$

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.

$$11x^{2}+4x-2=0$$

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

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

Pūrua 4.

$$x=\frac{-4±\sqrt{16-4\times 11\left(-2\right)}}{2\times 11}$$

Whakareatia -4 ki te 11.

$$x=\frac{-4±\sqrt{16-44\left(-2\right)}}{2\times 11}$$

Whakareatia -44 ki te -2.

$$x=\frac{-4±\sqrt{16+88}}{2\times 11}$$

Tāpiri 16 ki te 88.

$$x=\frac{-4±\sqrt{104}}{2\times 11}$$

Tuhia te pūtakerua o te 104.

$$x=\frac{-4±2\sqrt{26}}{2\times 11}$$

Whakareatia 2 ki te 11.

$$x=\frac{-4±2\sqrt{26}}{22}$$

Nā, me whakaoti te whārite x=\frac{-4±2\sqrt{26}}{22} ina he tāpiri te ±. Tāpiri -4 ki te 2\sqrt{26}.

$$x=\frac{2\sqrt{26}-4}{22}$$

Whakawehe -4+2\sqrt{26} ki te 22.

$$x=\frac{\sqrt{26}-2}{11}$$

Nā, me whakaoti te whārite x=\frac{-4±2\sqrt{26}}{22} ina he tango te ±. Tango 2\sqrt{26} mai i -4.

$$x=\frac{-2\sqrt{26}-4}{22}$$

Whakawehe -4-2\sqrt{26} ki te 22.

$$x=\frac{-\sqrt{26}-2}{11}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{26}-2}{11} x=\frac{-\sqrt{26}-2}{11}$$

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.

$$11x^{2}+4x-2=0$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$11x^{2}+4x-2-\left(-2\right)=-\left(-2\right)$$

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

$$11x^{2}+4x=-\left(-2\right)$$

Tango -2 mai i 0.

$$11x^{2}+4x=2$$

Whakawehea ngā taha e rua ki te 11.

$$\frac{11x^{2}+4x}{11}=\frac{2}{11}$$

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

$$x^{2}+\frac{4}{11}x=\frac{2}{11}$$

Whakawehea te \frac{4}{11}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{2}{11}. Nā, tāpiria te pūrua o te \frac{2}{11} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{4}{11}x+\left(\frac{2}{11}\right)^{2}=\frac{2}{11}+\left(\frac{2}{11}\right)^{2}$$

Pūruatia \frac{2}{11} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{4}{11}x+\frac{4}{121}=\frac{2}{11}+\frac{4}{121}$$

Tāpiri \frac{2}{11} ki te \frac{4}{121} 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.

$$x^{2}+\frac{4}{11}x+\frac{4}{121}=\frac{26}{121}$$

Tauwehea x^{2}+\frac{4}{11}x+\frac{4}{121}. 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}.

$$\left(x+\frac{2}{11}\right)^{2}=\frac{26}{121}$$

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

$$\sqrt{\left(x+\frac{2}{11}\right)^{2}}=\sqrt{\frac{26}{121}}$$

Whakarūnātia.

$$x+\frac{2}{11}=\frac{\sqrt{26}}{11} x+\frac{2}{11}=-\frac{\sqrt{26}}{11}$$

Me tango \frac{2}{11} mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{26}-2}{11} x=\frac{-\sqrt{26}-2}{11}$$