Whakaoti i te 112=6x-1/2*75x^2

$$112 = 6 x - \frac { 1 } { 2 } \times 75 x ^ { 2 }$$

Whakaoti mō x (complex solution)

$$x=\frac{2\sqrt{4191}i}{75}+\frac{2}{25}\approx 0.08+1.726344886i$$

$$x=-\frac{2\sqrt{4191}i}{75}+\frac{2}{25}\approx 0.08-1.726344886i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakareatia te \frac{1}{2} ki te 75, ka \frac{75}{2}.

$$112=6x-\frac{75}{2}x^{2}$$

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

$$6x-\frac{75}{2}x^{2}=112$$

Tangohia te 112 mai i ngā taha e rua.

$$6x-\frac{75}{2}x^{2}-112=0$$

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.

$$-\frac{75}{2}x^{2}+6x-112=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -\frac{75}{2} mō a, 6 mō b, me -112 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\left(-\frac{75}{2}\right)\left(-112\right)}}{2\left(-\frac{75}{2}\right)}$$

Pūrua 6.

$$x=\frac{-6±\sqrt{36-4\left(-\frac{75}{2}\right)\left(-112\right)}}{2\left(-\frac{75}{2}\right)}$$

Whakareatia -4 ki te -\frac{75}{2}.

$$x=\frac{-6±\sqrt{36+150\left(-112\right)}}{2\left(-\frac{75}{2}\right)}$$

Whakareatia 150 ki te -112.

$$x=\frac{-6±\sqrt{36-16800}}{2\left(-\frac{75}{2}\right)}$$

Tāpiri 36 ki te -16800.

$$x=\frac{-6±\sqrt{-16764}}{2\left(-\frac{75}{2}\right)}$$

Tuhia te pūtakerua o te -16764.

$$x=\frac{-6±2\sqrt{4191}i}{2\left(-\frac{75}{2}\right)}$$

Whakareatia 2 ki te -\frac{75}{2}.

$$x=\frac{-6±2\sqrt{4191}i}{-75}$$

Nā, me whakaoti te whārite x=\frac{-6±2\sqrt{4191}i}{-75} ina he tāpiri te ±. Tāpiri -6 ki te 2i\sqrt{4191}.

$$x=\frac{-6+2\sqrt{4191}i}{-75}$$

Whakawehe -6+2i\sqrt{4191} ki te -75.

$$x=-\frac{2\sqrt{4191}i}{75}+\frac{2}{25}$$

Nā, me whakaoti te whārite x=\frac{-6±2\sqrt{4191}i}{-75} ina he tango te ±. Tango 2i\sqrt{4191} mai i -6.

$$x=\frac{-2\sqrt{4191}i-6}{-75}$$

Whakawehe -6-2i\sqrt{4191} ki te -75.

$$x=\frac{2\sqrt{4191}i}{75}+\frac{2}{25}$$

Kua oti te whārite te whakatau.

$$x=-\frac{2\sqrt{4191}i}{75}+\frac{2}{25} x=\frac{2\sqrt{4191}i}{75}+\frac{2}{25}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakareatia te \frac{1}{2} ki te 75, ka \frac{75}{2}.

$$112=6x-\frac{75}{2}x^{2}$$

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

$$6x-\frac{75}{2}x^{2}=112$$

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.

$$-\frac{75}{2}x^{2}+6x=112$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{75}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$\frac{-\frac{75}{2}x^{2}+6x}{-\frac{75}{2}}=\frac{112}{-\frac{75}{2}}$$

Mā te whakawehe ki te -\frac{75}{2} ka wetekia te whakareanga ki te -\frac{75}{2}.

$$x^{2}+\frac{6}{-\frac{75}{2}}x=\frac{112}{-\frac{75}{2}}$$

Whakawehe 6 ki te -\frac{75}{2} mā te whakarea 6 ki te tau huripoki o -\frac{75}{2}.

$$x^{2}-\frac{4}{25}x=\frac{112}{-\frac{75}{2}}$$

Whakawehe 112 ki te -\frac{75}{2} mā te whakarea 112 ki te tau huripoki o -\frac{75}{2}.

$$x^{2}-\frac{4}{25}x=-\frac{224}{75}$$

Whakawehea te -\frac{4}{25}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{2}{25}. Nā, tāpiria te pūrua o te -\frac{2}{25} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{4}{25}x+\left(-\frac{2}{25}\right)^{2}=-\frac{224}{75}+\left(-\frac{2}{25}\right)^{2}$$

Pūruatia -\frac{2}{25} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{4}{25}x+\frac{4}{625}=-\frac{224}{75}+\frac{4}{625}$$

Tāpiri -\frac{224}{75} ki te \frac{4}{625} 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.

$$x^{2}-\frac{4}{25}x+\frac{4}{625}=-\frac{5588}{1875}$$

Tauwehea x^{2}-\frac{4}{25}x+\frac{4}{625}. 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}.

$$\left(x-\frac{2}{25}\right)^{2}=-\frac{5588}{1875}$$

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

$$\sqrt{\left(x-\frac{2}{25}\right)^{2}}=\sqrt{-\frac{5588}{1875}}$$

Whakarūnātia.

$$x-\frac{2}{25}=\frac{2\sqrt{4191}i}{75} x-\frac{2}{25}=-\frac{2\sqrt{4191}i}{75}$$

Me tāpiri \frac{2}{25} ki ngā taha e rua o te whārite.

$$x=\frac{2\sqrt{4191}i}{75}+\frac{2}{25} x=-\frac{2\sqrt{4191}i}{75}+\frac{2}{25}$$