Whakaoti i te 18=-1/5x^2-12x+32

$$18 = - \frac { 1 } { 5 } x ^ { 2 } - 12 x + 32$$

Whakaoti mō x (complex solution)

$$x=\sqrt{970}-30\approx 1.144823005$$

$$x=-\left(\sqrt{970}+30\right)\approx -61.144823005$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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

$$-\frac{1}{5}x^{2}-12x+32=18$$

Tangohia te 18 mai i ngā taha e rua.

$$-\frac{1}{5}x^{2}-12x+32-18=0$$

Tangohia te 18 i te 32, ka 14.

$$-\frac{1}{5}x^{2}-12x+14=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -\frac{1}{5} mō a, -12 mō b, me 14 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\left(-\frac{1}{5}\right)\times 14}}{2\left(-\frac{1}{5}\right)}$$

Pūrua -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\left(-\frac{1}{5}\right)\times 14}}{2\left(-\frac{1}{5}\right)}$$

Whakareatia -4 ki te -\frac{1}{5}.

$$x=\frac{-\left(-12\right)±\sqrt{144+\frac{4}{5}\times 14}}{2\left(-\frac{1}{5}\right)}$$

Whakareatia \frac{4}{5} ki te 14.

$$x=\frac{-\left(-12\right)±\sqrt{144+\frac{56}{5}}}{2\left(-\frac{1}{5}\right)}$$

Tāpiri 144 ki te \frac{56}{5}.

$$x=\frac{-\left(-12\right)±\sqrt{\frac{776}{5}}}{2\left(-\frac{1}{5}\right)}$$

Tuhia te pūtakerua o te \frac{776}{5}.

$$x=\frac{-\left(-12\right)±\frac{2\sqrt{970}}{5}}{2\left(-\frac{1}{5}\right)}$$

Ko te tauaro o -12 ko 12.

$$x=\frac{12±\frac{2\sqrt{970}}{5}}{2\left(-\frac{1}{5}\right)}$$

Whakareatia 2 ki te -\frac{1}{5}.

$$x=\frac{12±\frac{2\sqrt{970}}{5}}{-\frac{2}{5}}$$

Nā, me whakaoti te whārite x=\frac{12±\frac{2\sqrt{970}}{5}}{-\frac{2}{5}} ina he tāpiri te ±. Tāpiri 12 ki te \frac{2\sqrt{970}}{5}.

$$x=\frac{\frac{2\sqrt{970}}{5}+12}{-\frac{2}{5}}$$

Whakawehe 12+\frac{2\sqrt{970}}{5} ki te -\frac{2}{5} mā te whakarea 12+\frac{2\sqrt{970}}{5} ki te tau huripoki o -\frac{2}{5}.

$$x=-\left(\sqrt{970}+30\right)$$

Nā, me whakaoti te whārite x=\frac{12±\frac{2\sqrt{970}}{5}}{-\frac{2}{5}} ina he tango te ±. Tango \frac{2\sqrt{970}}{5} mai i 12.

$$x=\frac{-\frac{2\sqrt{970}}{5}+12}{-\frac{2}{5}}$$

Whakawehe 12-\frac{2\sqrt{970}}{5} ki te -\frac{2}{5} mā te whakarea 12-\frac{2\sqrt{970}}{5} ki te tau huripoki o -\frac{2}{5}.

$$x=\sqrt{970}-30$$

Kua oti te whārite te whakatau.

$$x=-\left(\sqrt{970}+30\right) x=\sqrt{970}-30$$

Ngā Upane mō te Whakaoti i te Pūrua

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

$$-\frac{1}{5}x^{2}-12x+32=18$$

Tangohia te 32 mai i ngā taha e rua.

$$-\frac{1}{5}x^{2}-12x=18-32$$

Tangohia te 32 i te 18, ka -14.

$$-\frac{1}{5}x^{2}-12x=-14$$

Me whakarea ngā taha e rua ki te -5.

$$\frac{-\frac{1}{5}x^{2}-12x}{-\frac{1}{5}}=-\frac{14}{-\frac{1}{5}}$$

Mā te whakawehe ki te -\frac{1}{5} ka wetekia te whakareanga ki te -\frac{1}{5}.

$$x^{2}+\left(-\frac{12}{-\frac{1}{5}}\right)x=-\frac{14}{-\frac{1}{5}}$$

Whakawehe -12 ki te -\frac{1}{5} mā te whakarea -12 ki te tau huripoki o -\frac{1}{5}.

$$x^{2}+60x=-\frac{14}{-\frac{1}{5}}$$

Whakawehe -14 ki te -\frac{1}{5} mā te whakarea -14 ki te tau huripoki o -\frac{1}{5}.

$$x^{2}+60x=70$$

Whakawehea te 60, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 30. Nā, tāpiria te pūrua o te 30 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+60x+30^{2}=70+30^{2}$$

Pūrua 30.

$$x^{2}+60x+900=70+900$$

Tāpiri 70 ki te 900.

$$x^{2}+60x+900=970$$

Tauwehea x^{2}+60x+900. 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}.

$$\left(x+30\right)^{2}=970$$

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

$$\sqrt{\left(x+30\right)^{2}}=\sqrt{970}$$

Whakarūnātia.

$$x+30=\sqrt{970} x+30=-\sqrt{970}$$

Me tango 30 mai i ngā taha e rua o te whārite.

$$x=\sqrt{970}-30 x=-\sqrt{970}-30$$

Whakaoti mō x

$$x=\sqrt{970}-30\approx 1.144823005$$

$$x=-\sqrt{970}-30\approx -61.144823005$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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

$$-\frac{1}{5}x^{2}-12x+32=18$$

Tangohia te 18 mai i ngā taha e rua.

$$-\frac{1}{5}x^{2}-12x+32-18=0$$

Tangohia te 18 i te 32, ka 14.

$$-\frac{1}{5}x^{2}-12x+14=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -\frac{1}{5} mō a, -12 mō b, me 14 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\left(-\frac{1}{5}\right)\times 14}}{2\left(-\frac{1}{5}\right)}$$

Pūrua -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\left(-\frac{1}{5}\right)\times 14}}{2\left(-\frac{1}{5}\right)}$$

Whakareatia -4 ki te -\frac{1}{5}.

$$x=\frac{-\left(-12\right)±\sqrt{144+\frac{4}{5}\times 14}}{2\left(-\frac{1}{5}\right)}$$

Whakareatia \frac{4}{5} ki te 14.

$$x=\frac{-\left(-12\right)±\sqrt{144+\frac{56}{5}}}{2\left(-\frac{1}{5}\right)}$$

Tāpiri 144 ki te \frac{56}{5}.

$$x=\frac{-\left(-12\right)±\sqrt{\frac{776}{5}}}{2\left(-\frac{1}{5}\right)}$$

Tuhia te pūtakerua o te \frac{776}{5}.

$$x=\frac{-\left(-12\right)±\frac{2\sqrt{970}}{5}}{2\left(-\frac{1}{5}\right)}$$

Ko te tauaro o -12 ko 12.

$$x=\frac{12±\frac{2\sqrt{970}}{5}}{2\left(-\frac{1}{5}\right)}$$

Whakareatia 2 ki te -\frac{1}{5}.

$$x=\frac{12±\frac{2\sqrt{970}}{5}}{-\frac{2}{5}}$$

Nā, me whakaoti te whārite x=\frac{12±\frac{2\sqrt{970}}{5}}{-\frac{2}{5}} ina he tāpiri te ±. Tāpiri 12 ki te \frac{2\sqrt{970}}{5}.

$$x=\frac{\frac{2\sqrt{970}}{5}+12}{-\frac{2}{5}}$$

Whakawehe 12+\frac{2\sqrt{970}}{5} ki te -\frac{2}{5} mā te whakarea 12+\frac{2\sqrt{970}}{5} ki te tau huripoki o -\frac{2}{5}.

$$x=-\left(\sqrt{970}+30\right)$$

Nā, me whakaoti te whārite x=\frac{12±\frac{2\sqrt{970}}{5}}{-\frac{2}{5}} ina he tango te ±. Tango \frac{2\sqrt{970}}{5} mai i 12.

$$x=\frac{-\frac{2\sqrt{970}}{5}+12}{-\frac{2}{5}}$$

Whakawehe 12-\frac{2\sqrt{970}}{5} ki te -\frac{2}{5} mā te whakarea 12-\frac{2\sqrt{970}}{5} ki te tau huripoki o -\frac{2}{5}.

$$x=\sqrt{970}-30$$

Kua oti te whārite te whakatau.

$$x=-\left(\sqrt{970}+30\right) x=\sqrt{970}-30$$

Ngā Upane mō te Whakaoti i te Pūrua

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

$$-\frac{1}{5}x^{2}-12x+32=18$$

Tangohia te 32 mai i ngā taha e rua.

$$-\frac{1}{5}x^{2}-12x=18-32$$

Tangohia te 32 i te 18, ka -14.

$$-\frac{1}{5}x^{2}-12x=-14$$

Me whakarea ngā taha e rua ki te -5.

$$\frac{-\frac{1}{5}x^{2}-12x}{-\frac{1}{5}}=-\frac{14}{-\frac{1}{5}}$$

Mā te whakawehe ki te -\frac{1}{5} ka wetekia te whakareanga ki te -\frac{1}{5}.

$$x^{2}+\left(-\frac{12}{-\frac{1}{5}}\right)x=-\frac{14}{-\frac{1}{5}}$$

Whakawehe -12 ki te -\frac{1}{5} mā te whakarea -12 ki te tau huripoki o -\frac{1}{5}.

$$x^{2}+60x=-\frac{14}{-\frac{1}{5}}$$

Whakawehe -14 ki te -\frac{1}{5} mā te whakarea -14 ki te tau huripoki o -\frac{1}{5}.

$$x^{2}+60x=70$$

Whakawehea te 60, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 30. Nā, tāpiria te pūrua o te 30 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+60x+30^{2}=70+30^{2}$$

Pūrua 30.

$$x^{2}+60x+900=70+900$$

Tāpiri 70 ki te 900.

$$x^{2}+60x+900=970$$

Tauwehea x^{2}+60x+900. 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}.

$$\left(x+30\right)^{2}=970$$

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

$$\sqrt{\left(x+30\right)^{2}}=\sqrt{970}$$

Whakarūnātia.

$$x+30=\sqrt{970} x+30=-\sqrt{970}$$

Me tango 30 mai i ngā taha e rua o te whārite.

$$x=\sqrt{970}-30 x=-\sqrt{970}-30$$