Whakaoti i te 30=-8x-49x^2

$$30=-8x-49 { x }^{ 2 }$$

Whakaoti mō x (complex solution)

$$x=\frac{-4+\sqrt{1454}i}{49}\approx -0.081632653+0.778190856i$$

$$x=\frac{-\sqrt{1454}i-4}{49}\approx -0.081632653-0.778190856i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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

$$-8x-49x^{2}=30$$

Tangohia te 30 mai i ngā taha e rua.

$$-8x-49x^{2}-30=0$$

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.

$$-49x^{2}-8x-30=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -49 mō a, -8 mō b, me -30 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-49\right)\left(-30\right)}}{2\left(-49\right)}$$

Pūrua -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\left(-49\right)\left(-30\right)}}{2\left(-49\right)}$$

Whakareatia -4 ki te -49.

$$x=\frac{-\left(-8\right)±\sqrt{64+196\left(-30\right)}}{2\left(-49\right)}$$

Whakareatia 196 ki te -30.

$$x=\frac{-\left(-8\right)±\sqrt{64-5880}}{2\left(-49\right)}$$

Tāpiri 64 ki te -5880.

$$x=\frac{-\left(-8\right)±\sqrt{-5816}}{2\left(-49\right)}$$

Tuhia te pūtakerua o te -5816.

$$x=\frac{-\left(-8\right)±2\sqrt{1454}i}{2\left(-49\right)}$$

Ko te tauaro o -8 ko 8.

$$x=\frac{8±2\sqrt{1454}i}{2\left(-49\right)}$$

Whakareatia 2 ki te -49.

$$x=\frac{8±2\sqrt{1454}i}{-98}$$

Nā, me whakaoti te whārite x=\frac{8±2\sqrt{1454}i}{-98} ina he tāpiri te ±. Tāpiri 8 ki te 2i\sqrt{1454}.

$$x=\frac{8+2\sqrt{1454}i}{-98}$$

Whakawehe 8+2i\sqrt{1454} ki te -98.

$$x=\frac{-\sqrt{1454}i-4}{49}$$

Nā, me whakaoti te whārite x=\frac{8±2\sqrt{1454}i}{-98} ina he tango te ±. Tango 2i\sqrt{1454} mai i 8.

$$x=\frac{-2\sqrt{1454}i+8}{-98}$$

Whakawehe 8-2i\sqrt{1454} ki te -98.

$$x=\frac{-4+\sqrt{1454}i}{49}$$

Kua oti te whārite te whakatau.

$$x=\frac{-\sqrt{1454}i-4}{49} x=\frac{-4+\sqrt{1454}i}{49}$$

Ngā Upane mō te Whakaoti i te Pūrua

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

$$-8x-49x^{2}=30$$

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.

$$-49x^{2}-8x=30$$

Whakawehea ngā taha e rua ki te -49.

$$\frac{-49x^{2}-8x}{-49}=\frac{30}{-49}$$

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

$$x^{2}+\left(-\frac{8}{-49}\right)x=\frac{30}{-49}$$

Whakawehe -8 ki te -49.

$$x^{2}+\frac{8}{49}x=\frac{30}{-49}$$

Whakawehe 30 ki te -49.

$$x^{2}+\frac{8}{49}x=-\frac{30}{49}$$

Whakawehea te \frac{8}{49}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{4}{49}. Nā, tāpiria te pūrua o te \frac{4}{49} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{8}{49}x+\left(\frac{4}{49}\right)^{2}=-\frac{30}{49}+\left(\frac{4}{49}\right)^{2}$$

Pūruatia \frac{4}{49} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{8}{49}x+\frac{16}{2401}=-\frac{30}{49}+\frac{16}{2401}$$

Tāpiri -\frac{30}{49} ki te \frac{16}{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.

$$x^{2}+\frac{8}{49}x+\frac{16}{2401}=-\frac{1454}{2401}$$

Tauwehea x^{2}+\frac{8}{49}x+\frac{16}{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}.

$$\left(x+\frac{4}{49}\right)^{2}=-\frac{1454}{2401}$$

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

$$\sqrt{\left(x+\frac{4}{49}\right)^{2}}=\sqrt{-\frac{1454}{2401}}$$

Whakarūnātia.

$$x+\frac{4}{49}=\frac{\sqrt{1454}i}{49} x+\frac{4}{49}=-\frac{\sqrt{1454}i}{49}$$

Me tango \frac{4}{49} mai i ngā taha e rua o te whārite.

$$x=\frac{-4+\sqrt{1454}i}{49} x=\frac{-\sqrt{1454}i-4}{49}$$