Whakaoti i te (50+x-40)(500-10x)=8000

$$(50+x-40)(500-10x)=8000$$

Whakaoti mō x

$$x=10$$

$$x=30$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 40 i te 50, ka 10.

$$\left(10+x\right)\left(500-10x\right)=8000$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 10+x ki te 500-10x ka whakakotahi i ngā kupu rite.

$$5000+400x-10x^{2}=8000$$

Tangohia te 8000 mai i ngā taha e rua.

$$5000+400x-10x^{2}-8000=0$$

Tangohia te 8000 i te 5000, ka -3000.

$$-3000+400x-10x^{2}=0$$

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.

$$-10x^{2}+400x-3000=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -10 mō a, 400 mō b, me -3000 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-400±\sqrt{400^{2}-4\left(-10\right)\left(-3000\right)}}{2\left(-10\right)}$$

Pūrua 400.

$$x=\frac{-400±\sqrt{160000-4\left(-10\right)\left(-3000\right)}}{2\left(-10\right)}$$

Whakareatia -4 ki te -10.

$$x=\frac{-400±\sqrt{160000+40\left(-3000\right)}}{2\left(-10\right)}$$

Whakareatia 40 ki te -3000.

$$x=\frac{-400±\sqrt{160000-120000}}{2\left(-10\right)}$$

Tāpiri 160000 ki te -120000.

$$x=\frac{-400±\sqrt{40000}}{2\left(-10\right)}$$

Tuhia te pūtakerua o te 40000.

$$x=\frac{-400±200}{2\left(-10\right)}$$

Whakareatia 2 ki te -10.

$$x=\frac{-400±200}{-20}$$

Nā, me whakaoti te whārite x=\frac{-400±200}{-20} ina he tāpiri te ±. Tāpiri -400 ki te 200.

$$x=-\frac{200}{-20}$$

Whakawehe -200 ki te -20.

$$x=10$$

Nā, me whakaoti te whārite x=\frac{-400±200}{-20} ina he tango te ±. Tango 200 mai i -400.

$$x=-\frac{600}{-20}$$

Whakawehe -600 ki te -20.

$$x=30$$

Kua oti te whārite te whakatau.

$$x=10 x=30$$

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te 40 i te 50, ka 10.

$$\left(10+x\right)\left(500-10x\right)=8000$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 10+x ki te 500-10x ka whakakotahi i ngā kupu rite.

$$5000+400x-10x^{2}=8000$$

Tangohia te 5000 mai i ngā taha e rua.

$$400x-10x^{2}=8000-5000$$

Tangohia te 5000 i te 8000, ka 3000.

$$400x-10x^{2}=3000$$

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.

$$-10x^{2}+400x=3000$$

Whakawehea ngā taha e rua ki te -10.

$$\frac{-10x^{2}+400x}{-10}=\frac{3000}{-10}$$

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

$$x^{2}+\frac{400}{-10}x=\frac{3000}{-10}$$

Whakawehe 400 ki te -10.

$$x^{2}-40x=\frac{3000}{-10}$$

Whakawehe 3000 ki te -10.

$$x^{2}-40x=-300$$

Whakawehea te -40, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -20. Nā, tāpiria te pūrua o te -20 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-40x+\left(-20\right)^{2}=-300+\left(-20\right)^{2}$$

Pūrua -20.

$$x^{2}-40x+400=-300+400$$

Tāpiri -300 ki te 400.

$$x^{2}-40x+400=100$$

Tauwehea x^{2}-40x+400. 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}.

$$\left(x-20\right)^{2}=100$$

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

$$\sqrt{\left(x-20\right)^{2}}=\sqrt{100}$$

Whakarūnātia.

$$x-20=10 x-20=-10$$

Me tāpiri 20 ki ngā taha e rua o te whārite.

$$x=30 x=10$$