Whakaoti i te 11=1+20x-49x^2

$$11=1+20x-49 { x }^{ 2 }$$

Whakaoti mō x (complex solution)

$$x=\frac{-\sqrt{390}i+10}{49}\approx 0.204081633-0.403028932i$$

$$x=\frac{10+\sqrt{390}i}{49}\approx 0.204081633+0.403028932i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$1+20x-49x^{2}=11$$

Tangohia te 11 mai i ngā taha e rua.

$$1+20x-49x^{2}-11=0$$

Tangohia te 11 i te 1, ka -10.

$$-10+20x-49x^{2}=0$$

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.

$$-49x^{2}+20x-10=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -49 mō a, 20 mō b, me -10 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-20±\sqrt{20^{2}-4\left(-49\right)\left(-10\right)}}{2\left(-49\right)}$$

Pūrua 20.

$$x=\frac{-20±\sqrt{400-4\left(-49\right)\left(-10\right)}}{2\left(-49\right)}$$

Whakareatia -4 ki te -49.

$$x=\frac{-20±\sqrt{400+196\left(-10\right)}}{2\left(-49\right)}$$

Whakareatia 196 ki te -10.

$$x=\frac{-20±\sqrt{400-1960}}{2\left(-49\right)}$$

Tāpiri 400 ki te -1960.

$$x=\frac{-20±\sqrt{-1560}}{2\left(-49\right)}$$

Tuhia te pūtakerua o te -1560.

$$x=\frac{-20±2\sqrt{390}i}{2\left(-49\right)}$$

Whakareatia 2 ki te -49.

$$x=\frac{-20±2\sqrt{390}i}{-98}$$

Nā, me whakaoti te whārite x=\frac{-20±2\sqrt{390}i}{-98} ina he tāpiri te ±. Tāpiri -20 ki te 2i\sqrt{390}.

$$x=\frac{-20+2\sqrt{390}i}{-98}$$

Whakawehe -20+2i\sqrt{390} ki te -98.

$$x=\frac{-\sqrt{390}i+10}{49}$$

Nā, me whakaoti te whārite x=\frac{-20±2\sqrt{390}i}{-98} ina he tango te ±. Tango 2i\sqrt{390} mai i -20.

$$x=\frac{-2\sqrt{390}i-20}{-98}$$

Whakawehe -20-2i\sqrt{390} ki te -98.

$$x=\frac{10+\sqrt{390}i}{49}$$

Kua oti te whārite te whakatau.

$$x=\frac{-\sqrt{390}i+10}{49} x=\frac{10+\sqrt{390}i}{49}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$1+20x-49x^{2}=11$$

Tangohia te 1 mai i ngā taha e rua.

$$20x-49x^{2}=11-1$$

Tangohia te 1 i te 11, ka 10.

$$20x-49x^{2}=10$$

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.

$$-49x^{2}+20x=10$$

Whakawehea ngā taha e rua ki te -49.

$$\frac{-49x^{2}+20x}{-49}=\frac{10}{-49}$$

Mā te whakawehe ki te -49 ka wetekia te whakareanga ki te -49.

$$x^{2}+\frac{20}{-49}x=\frac{10}{-49}$$

Whakawehe 20 ki te -49.

$$x^{2}-\frac{20}{49}x=\frac{10}{-49}$$

Whakawehe 10 ki te -49.

$$x^{2}-\frac{20}{49}x=-\frac{10}{49}$$

Whakawehea te -\frac{20}{49}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{10}{49}. Nā, tāpiria te pūrua o te -\frac{10}{49} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{20}{49}x+\left(-\frac{10}{49}\right)^{2}=-\frac{10}{49}+\left(-\frac{10}{49}\right)^{2}$$

Pūruatia -\frac{10}{49} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{20}{49}x+\frac{100}{2401}=-\frac{10}{49}+\frac{100}{2401}$$

Tāpiri -\frac{10}{49} ki te \frac{100}{2401} 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.

$$x^{2}-\frac{20}{49}x+\frac{100}{2401}=-\frac{390}{2401}$$

Tauwehea x^{2}-\frac{20}{49}x+\frac{100}{2401}. 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}.

$$\left(x-\frac{10}{49}\right)^{2}=-\frac{390}{2401}$$

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

$$\sqrt{\left(x-\frac{10}{49}\right)^{2}}=\sqrt{-\frac{390}{2401}}$$

Whakarūnātia.

$$x-\frac{10}{49}=\frac{\sqrt{390}i}{49} x-\frac{10}{49}=-\frac{\sqrt{390}i}{49}$$

Me tāpiri \frac{10}{49} ki ngā taha e rua o te whārite.

$$x=\frac{10+\sqrt{390}i}{49} x=\frac{-\sqrt{390}i+10}{49}$$