Whakaoti i te 2100=-3p^2+132p

$$2100 = - 3 p ^ { 2 } + 132 p$$

Whakaoti mō p

$$p=22+6\sqrt{6}i\approx 22+14.696938457i$$

$$p=-6\sqrt{6}i+22\approx 22-14.696938457i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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

$$-3p^{2}+132p=2100$$

Tangohia te 2100 mai i ngā taha e rua.

$$-3p^{2}+132p-2100=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -3 mō a, 132 mō b, me -2100 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$p=\frac{-132±\sqrt{132^{2}-4\left(-3\right)\left(-2100\right)}}{2\left(-3\right)}$$

Pūrua 132.

$$p=\frac{-132±\sqrt{17424-4\left(-3\right)\left(-2100\right)}}{2\left(-3\right)}$$

Whakareatia -4 ki te -3.

$$p=\frac{-132±\sqrt{17424+12\left(-2100\right)}}{2\left(-3\right)}$$

Whakareatia 12 ki te -2100.

$$p=\frac{-132±\sqrt{17424-25200}}{2\left(-3\right)}$$

Tāpiri 17424 ki te -25200.

$$p=\frac{-132±\sqrt{-7776}}{2\left(-3\right)}$$

Tuhia te pūtakerua o te -7776.

$$p=\frac{-132±36\sqrt{6}i}{2\left(-3\right)}$$

Whakareatia 2 ki te -3.

$$p=\frac{-132±36\sqrt{6}i}{-6}$$

Nā, me whakaoti te whārite p=\frac{-132±36\sqrt{6}i}{-6} ina he tāpiri te ±. Tāpiri -132 ki te 36i\sqrt{6}.

$$p=\frac{-132+36\sqrt{6}i}{-6}$$

Whakawehe -132+36i\sqrt{6} ki te -6.

$$p=-6\sqrt{6}i+22$$

Nā, me whakaoti te whārite p=\frac{-132±36\sqrt{6}i}{-6} ina he tango te ±. Tango 36i\sqrt{6} mai i -132.

$$p=\frac{-36\sqrt{6}i-132}{-6}$$

Whakawehe -132-36i\sqrt{6} ki te -6.

$$p=22+6\sqrt{6}i$$

Kua oti te whārite te whakatau.

$$p=-6\sqrt{6}i+22 p=22+6\sqrt{6}i$$

Ngā Upane mō te Whakaoti i te Pūrua

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

$$-3p^{2}+132p=2100$$

Whakawehea ngā taha e rua ki te -3.

$$\frac{-3p^{2}+132p}{-3}=\frac{2100}{-3}$$

Mā te whakawehe ki te -3 ka wetekia te whakareanga ki te -3.

$$p^{2}+\frac{132}{-3}p=\frac{2100}{-3}$$

Whakawehe 132 ki te -3.

$$p^{2}-44p=\frac{2100}{-3}$$

Whakawehe 2100 ki te -3.

$$p^{2}-44p=-700$$

Whakawehea te -44, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -22. Nā, tāpiria te pūrua o te -22 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$p^{2}-44p+\left(-22\right)^{2}=-700+\left(-22\right)^{2}$$

Pūrua -22.

$$p^{2}-44p+484=-700+484$$

Tāpiri -700 ki te 484.

$$p^{2}-44p+484=-216$$

Tauwehea p^{2}-44p+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}.

$$\left(p-22\right)^{2}=-216$$

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

$$\sqrt{\left(p-22\right)^{2}}=\sqrt{-216}$$

Whakarūnātia.

$$p-22=6\sqrt{6}i p-22=-6\sqrt{6}i$$

Me tāpiri 22 ki ngā taha e rua o te whārite.

$$p=22+6\sqrt{6}i p=-6\sqrt{6}i+22$$