Whakaoti i te 96*20=(20-x)(126-2x)

$$96 \times 20 = ( 20 - x ) ( 126 - 2 x )$$

Whakaoti mō x

$$x = \frac{\sqrt{5689} + 83}{2} \approx 79.212729946$$

$$x = \frac{83 - \sqrt{5689}}{2} \approx 3.787270054$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakareatia te 96 ki te 20, ka 1920.

$$1920=\left(20-x\right)\left(126-2x\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 20-x ki te 126-2x ka whakakotahi i ngā kupu rite.

$$1920=2520-166x+2x^{2}$$

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

$$2520-166x+2x^{2}=1920$$

Tangohia te 1920 mai i ngā taha e rua.

$$2520-166x+2x^{2}-1920=0$$

Tangohia te 1920 i te 2520, ka 600.

$$600-166x+2x^{2}=0$$

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.

$$2x^{2}-166x+600=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, -166 mō b, me 600 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-166\right)±\sqrt{\left(-166\right)^{2}-4\times 2\times 600}}{2\times 2}$$

Pūrua -166.

$$x=\frac{-\left(-166\right)±\sqrt{27556-4\times 2\times 600}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-\left(-166\right)±\sqrt{27556-8\times 600}}{2\times 2}$$

Whakareatia -8 ki te 600.

$$x=\frac{-\left(-166\right)±\sqrt{27556-4800}}{2\times 2}$$

Tāpiri 27556 ki te -4800.

$$x=\frac{-\left(-166\right)±\sqrt{22756}}{2\times 2}$$

Tuhia te pūtakerua o te 22756.

$$x=\frac{-\left(-166\right)±2\sqrt{5689}}{2\times 2}$$

Ko te tauaro o -166 ko 166.

$$x=\frac{166±2\sqrt{5689}}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{166±2\sqrt{5689}}{4}$$

Nā, me whakaoti te whārite x=\frac{166±2\sqrt{5689}}{4} ina he tāpiri te ±. Tāpiri 166 ki te 2\sqrt{5689}.

$$x=\frac{2\sqrt{5689}+166}{4}$$

Whakawehe 166+2\sqrt{5689} ki te 4.

$$x=\frac{\sqrt{5689}+83}{2}$$

Nā, me whakaoti te whārite x=\frac{166±2\sqrt{5689}}{4} ina he tango te ±. Tango 2\sqrt{5689} mai i 166.

$$x=\frac{166-2\sqrt{5689}}{4}$$

Whakawehe 166-2\sqrt{5689} ki te 4.

$$x=\frac{83-\sqrt{5689}}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{5689}+83}{2} x=\frac{83-\sqrt{5689}}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakareatia te 96 ki te 20, ka 1920.

$$1920=\left(20-x\right)\left(126-2x\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 20-x ki te 126-2x ka whakakotahi i ngā kupu rite.

$$1920=2520-166x+2x^{2}$$

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

$$2520-166x+2x^{2}=1920$$

Tangohia te 2520 mai i ngā taha e rua.

$$-166x+2x^{2}=1920-2520$$

Tangohia te 2520 i te 1920, ka -600.

$$-166x+2x^{2}=-600$$

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.

$$2x^{2}-166x=-600$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}-166x}{2}=-\frac{600}{2}$$

Mā te whakawehe ki te 2 ka wetekia te whakareanga ki te 2.

$$x^{2}+\left(-\frac{166}{2}\right)x=-\frac{600}{2}$$

Whakawehe -166 ki te 2.

$$x^{2}-83x=-\frac{600}{2}$$

Whakawehe -600 ki te 2.

$$x^{2}-83x=-300$$

Whakawehea te -83, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{83}{2}. Nā, tāpiria te pūrua o te -\frac{83}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-83x+\left(-\frac{83}{2}\right)^{2}=-300+\left(-\frac{83}{2}\right)^{2}$$

Pūruatia -\frac{83}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-83x+\frac{6889}{4}=-300+\frac{6889}{4}$$

Tāpiri -300 ki te \frac{6889}{4}.

$$x^{2}-83x+\frac{6889}{4}=\frac{5689}{4}$$

Tauwehea x^{2}-83x+\frac{6889}{4}. 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}.

$$\left(x-\frac{83}{2}\right)^{2}=\frac{5689}{4}$$

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

$$\sqrt{\left(x-\frac{83}{2}\right)^{2}}=\sqrt{\frac{5689}{4}}$$

Whakarūnātia.

$$x-\frac{83}{2}=\frac{\sqrt{5689}}{2} x-\frac{83}{2}=-\frac{\sqrt{5689}}{2}$$

Me tāpiri \frac{83}{2} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{5689}+83}{2} x=\frac{83-\sqrt{5689}}{2}$$