Whakaoti i te 10000=(x-30)(1000-10x)

$$10000 = ( x - 30 ) ( 1000 - 10 x )$$

Whakaoti mō x

$$x=50$$

$$x=80$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te x-30 ki te 1000-10x ka whakakotahi i ngā kupu rite.

$$10000=1300x-10x^{2}-30000$$

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

$$1300x-10x^{2}-30000=10000$$

Tangohia te 10000 mai i ngā taha e rua.

$$1300x-10x^{2}-30000-10000=0$$

Tangohia te 10000 i te -30000, ka -40000.

$$1300x-10x^{2}-40000=0$$

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.

$$-10x^{2}+1300x-40000=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -10 mō a, 1300 mō b, me -40000 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1300±\sqrt{1300^{2}-4\left(-10\right)\left(-40000\right)}}{2\left(-10\right)}$$

Pūrua 1300.

$$x=\frac{-1300±\sqrt{1690000-4\left(-10\right)\left(-40000\right)}}{2\left(-10\right)}$$

Whakareatia -4 ki te -10.

$$x=\frac{-1300±\sqrt{1690000+40\left(-40000\right)}}{2\left(-10\right)}$$

Whakareatia 40 ki te -40000.

$$x=\frac{-1300±\sqrt{1690000-1600000}}{2\left(-10\right)}$$

Tāpiri 1690000 ki te -1600000.

$$x=\frac{-1300±\sqrt{90000}}{2\left(-10\right)}$$

Tuhia te pūtakerua o te 90000.

$$x=\frac{-1300±300}{2\left(-10\right)}$$

Whakareatia 2 ki te -10.

$$x=\frac{-1300±300}{-20}$$

Nā, me whakaoti te whārite x=\frac{-1300±300}{-20} ina he tāpiri te ±. Tāpiri -1300 ki te 300.

$$x=-\frac{1000}{-20}$$

Whakawehe -1000 ki te -20.

$$x=50$$

Nā, me whakaoti te whārite x=\frac{-1300±300}{-20} ina he tango te ±. Tango 300 mai i -1300.

$$x=-\frac{1600}{-20}$$

Whakawehe -1600 ki te -20.

$$x=80$$

Kua oti te whārite te whakatau.

$$x=50 x=80$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te x-30 ki te 1000-10x ka whakakotahi i ngā kupu rite.

$$10000=1300x-10x^{2}-30000$$

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

$$1300x-10x^{2}-30000=10000$$

Me tāpiri te 30000 ki ngā taha e rua.

$$1300x-10x^{2}=10000+30000$$

Tāpirihia te 10000 ki te 30000, ka 40000.

$$1300x-10x^{2}=40000$$

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.

$$-10x^{2}+1300x=40000$$

Whakawehea ngā taha e rua ki te -10.

$$\frac{-10x^{2}+1300x}{-10}=\frac{40000}{-10}$$

Mā te whakawehe ki te -10 ka wetekia te whakareanga ki te -10.

$$x^{2}+\frac{1300}{-10}x=\frac{40000}{-10}$$

Whakawehe 1300 ki te -10.

$$x^{2}-130x=\frac{40000}{-10}$$

Whakawehe 40000 ki te -10.

$$x^{2}-130x=-4000$$

Whakawehea te -130, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -65. Nā, tāpiria te pūrua o te -65 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-130x+\left(-65\right)^{2}=-4000+\left(-65\right)^{2}$$

Pūrua -65.

$$x^{2}-130x+4225=-4000+4225$$

Tāpiri -4000 ki te 4225.

$$x^{2}-130x+4225=225$$

Tauwehea x^{2}-130x+4225. 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}.

$$\left(x-65\right)^{2}=225$$

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

$$\sqrt{\left(x-65\right)^{2}}=\sqrt{225}$$

Whakarūnātia.

$$x-65=15 x-65=-15$$

Me tāpiri 65 ki ngā taha e rua o te whārite.

$$x=80 x=50$$