Whakaoti i te 30=-8x-4.9x^2

$$30=-8x-4.9 { x }^{ 2 }$$

Whakaoti mō x (complex solution)

$$x=\frac{-40+i\times 10\sqrt{131}}{49}\approx -0.816326531+2.335821049i$$

$$x=\frac{-i\times 10\sqrt{131}-40}{49}\approx -0.816326531-2.335821049i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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

$$-8x-4.9x^{2}=30$$

Tangohia te 30 mai i ngā taha e rua.

$$-8x-4.9x^{2}-30=0$$

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.

$$-4.9x^{2}-8x-30=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -4.9 mō a, -8 mō b, me -30 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-4.9\right)\left(-30\right)}}{2\left(-4.9\right)}$$

Pūrua -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\left(-4.9\right)\left(-30\right)}}{2\left(-4.9\right)}$$

Whakareatia -4 ki te -4.9.

$$x=\frac{-\left(-8\right)±\sqrt{64+19.6\left(-30\right)}}{2\left(-4.9\right)}$$

Whakareatia 19.6 ki te -30.

$$x=\frac{-\left(-8\right)±\sqrt{64-588}}{2\left(-4.9\right)}$$

Tāpiri 64 ki te -588.

$$x=\frac{-\left(-8\right)±\sqrt{-524}}{2\left(-4.9\right)}$$

Tuhia te pūtakerua o te -524.

$$x=\frac{-\left(-8\right)±2\sqrt{131}i}{2\left(-4.9\right)}$$

Ko te tauaro o -8 ko 8.

$$x=\frac{8±2\sqrt{131}i}{2\left(-4.9\right)}$$

Whakareatia 2 ki te -4.9.

$$x=\frac{8±2\sqrt{131}i}{-9.8}$$

Nā, me whakaoti te whārite x=\frac{8±2\sqrt{131}i}{-9.8} ina he tāpiri te ±. Tāpiri 8 ki te 2i\sqrt{131}.

$$x=\frac{8+2\sqrt{131}i}{-9.8}$$

Whakawehe 8+2i\sqrt{131} ki te -9.8 mā te whakarea 8+2i\sqrt{131} ki te tau huripoki o -9.8.

$$x=\frac{-10\sqrt{131}i-40}{49}$$

Nā, me whakaoti te whārite x=\frac{8±2\sqrt{131}i}{-9.8} ina he tango te ±. Tango 2i\sqrt{131} mai i 8.

$$x=\frac{-2\sqrt{131}i+8}{-9.8}$$

Whakawehe 8-2i\sqrt{131} ki te -9.8 mā te whakarea 8-2i\sqrt{131} ki te tau huripoki o -9.8.

$$x=\frac{-40+10\sqrt{131}i}{49}$$

Kua oti te whārite te whakatau.

$$x=\frac{-10\sqrt{131}i-40}{49} x=\frac{-40+10\sqrt{131}i}{49}$$

Ngā Upane mō te Whakaoti i te Pūrua

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

$$-8x-4.9x^{2}=30$$

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.

$$-4.9x^{2}-8x=30$$

Whakawehea ngā taha e rua o te whārite ki te -4.9, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$\frac{-4.9x^{2}-8x}{-4.9}=\frac{30}{-4.9}$$

Mā te whakawehe ki te -4.9 ka wetekia te whakareanga ki te -4.9.

$$x^{2}+\left(-\frac{8}{-4.9}\right)x=\frac{30}{-4.9}$$

Whakawehe -8 ki te -4.9 mā te whakarea -8 ki te tau huripoki o -4.9.

$$x^{2}+\frac{80}{49}x=\frac{30}{-4.9}$$

Whakawehe 30 ki te -4.9 mā te whakarea 30 ki te tau huripoki o -4.9.

$$x^{2}+\frac{80}{49}x=-\frac{300}{49}$$

Whakawehea te \frac{80}{49}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{40}{49}. Nā, tāpiria te pūrua o te \frac{40}{49} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{80}{49}x+\frac{40}{49}^{2}=-\frac{300}{49}+\frac{40}{49}^{2}$$

Pūruatia \frac{40}{49} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{80}{49}x+\frac{1600}{2401}=-\frac{300}{49}+\frac{1600}{2401}$$

Tāpiri -\frac{300}{49} ki te \frac{1600}{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.

$$x^{2}+\frac{80}{49}x+\frac{1600}{2401}=-\frac{13100}{2401}$$

Tauwehea x^{2}+\frac{80}{49}x+\frac{1600}{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}.

$$\left(x+\frac{40}{49}\right)^{2}=-\frac{13100}{2401}$$

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

$$\sqrt{\left(x+\frac{40}{49}\right)^{2}}=\sqrt{-\frac{13100}{2401}}$$

Whakarūnātia.

$$x+\frac{40}{49}=\frac{10\sqrt{131}i}{49} x+\frac{40}{49}=-\frac{10\sqrt{131}i}{49}$$

Me tango \frac{40}{49} mai i ngā taha e rua o te whārite.

$$x=\frac{-40+10\sqrt{131}i}{49} x=\frac{-10\sqrt{131}i-40}{49}$$