Whakaoti i te 200=2x(20-x)

$$200 = 2 x ( 20 - x )$$

Whakaoti mō x

$$x=10$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 2x ki te 20-x.

$$200=40x-2x^{2}$$

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

$$40x-2x^{2}=200$$

Tangohia te 200 mai i ngā taha e rua.

$$40x-2x^{2}-200=0$$

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.

$$-2x^{2}+40x-200=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -2 mō a, 40 mō b, me -200 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-40±\sqrt{40^{2}-4\left(-2\right)\left(-200\right)}}{2\left(-2\right)}$$

Pūrua 40.

$$x=\frac{-40±\sqrt{1600-4\left(-2\right)\left(-200\right)}}{2\left(-2\right)}$$

Whakareatia -4 ki te -2.

$$x=\frac{-40±\sqrt{1600+8\left(-200\right)}}{2\left(-2\right)}$$

Whakareatia 8 ki te -200.

$$x=\frac{-40±\sqrt{1600-1600}}{2\left(-2\right)}$$

Tāpiri 1600 ki te -1600.

$$x=\frac{-40±\sqrt{0}}{2\left(-2\right)}$$

Tuhia te pūtakerua o te 0.

$$x=-\frac{40}{2\left(-2\right)}$$

Whakareatia 2 ki te -2.

$$x=-\frac{40}{-4}$$

Whakawehe -40 ki te -4.

$$x=10$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 2x ki te 20-x.

$$200=40x-2x^{2}$$

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

$$40x-2x^{2}=200$$

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.

$$-2x^{2}+40x=200$$

Whakawehea ngā taha e rua ki te -2.

$$\frac{-2x^{2}+40x}{-2}=\frac{200}{-2}$$

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

$$x^{2}+\frac{40}{-2}x=\frac{200}{-2}$$

Whakawehe 40 ki te -2.

$$x^{2}-20x=\frac{200}{-2}$$

Whakawehe 200 ki te -2.

$$x^{2}-20x=-100$$

Whakawehea te -20, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -10. Nā, tāpiria te pūrua o te -10 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-20x+\left(-10\right)^{2}=-100+\left(-10\right)^{2}$$

Pūrua -10.

$$x^{2}-20x+100=-100+100$$

Tāpiri -100 ki te 100.

$$x^{2}-20x+100=0$$

Tauwehea x^{2}-20x+100. 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}.

$$\left(x-10\right)^{2}=0$$

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

$$\sqrt{\left(x-10\right)^{2}}=\sqrt{0}$$

Whakarūnātia.

$$x-10=0 x-10=0$$

Me tāpiri 10 ki ngā taha e rua o te whārite.

$$x=10 x=10$$

Kua oti te whārite te whakatau. He ōrite ngā whakatau.

$$x=10$$