Whakaoti i te 450=x(100-2x)

$$450=x(100-2x)$$

Whakaoti mō x

$$x=5$$

$$x=45$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te 100-2x.

$$450=100x-2x^{2}$$

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

$$100x-2x^{2}=450$$

Tangohia te 450 mai i ngā taha e rua.

$$100x-2x^{2}-450=0$$

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.

$$-2x^{2}+100x-450=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -2 mō a, 100 mō b, me -450 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-100±\sqrt{100^{2}-4\left(-2\right)\left(-450\right)}}{2\left(-2\right)}$$

Pūrua 100.

$$x=\frac{-100±\sqrt{10000-4\left(-2\right)\left(-450\right)}}{2\left(-2\right)}$$

Whakareatia -4 ki te -2.

$$x=\frac{-100±\sqrt{10000+8\left(-450\right)}}{2\left(-2\right)}$$

Whakareatia 8 ki te -450.

$$x=\frac{-100±\sqrt{10000-3600}}{2\left(-2\right)}$$

Tāpiri 10000 ki te -3600.

$$x=\frac{-100±\sqrt{6400}}{2\left(-2\right)}$$

Tuhia te pūtakerua o te 6400.

$$x=\frac{-100±80}{2\left(-2\right)}$$

Whakareatia 2 ki te -2.

$$x=\frac{-100±80}{-4}$$

Nā, me whakaoti te whārite x=\frac{-100±80}{-4} ina he tāpiri te ±. Tāpiri -100 ki te 80.

$$x=-\frac{20}{-4}$$

Whakawehe -20 ki te -4.

$$x=5$$

Nā, me whakaoti te whārite x=\frac{-100±80}{-4} ina he tango te ±. Tango 80 mai i -100.

$$x=-\frac{180}{-4}$$

Whakawehe -180 ki te -4.

$$x=45$$

Kua oti te whārite te whakatau.

$$x=5 x=45$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te 100-2x.

$$450=100x-2x^{2}$$

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

$$100x-2x^{2}=450$$

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.

$$-2x^{2}+100x=450$$

Whakawehea ngā taha e rua ki te -2.

$$\frac{-2x^{2}+100x}{-2}=\frac{450}{-2}$$

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

$$x^{2}+\frac{100}{-2}x=\frac{450}{-2}$$

Whakawehe 100 ki te -2.

$$x^{2}-50x=\frac{450}{-2}$$

Whakawehe 450 ki te -2.

$$x^{2}-50x=-225$$

Whakawehea te -50, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -25. Nā, tāpiria te pūrua o te -25 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-50x+\left(-25\right)^{2}=-225+\left(-25\right)^{2}$$

Pūrua -25.

$$x^{2}-50x+625=-225+625$$

Tāpiri -225 ki te 625.

$$x^{2}-50x+625=400$$

Tauwehea x^{2}-50x+625. 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}.

$$\left(x-25\right)^{2}=400$$

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

$$\sqrt{\left(x-25\right)^{2}}=\sqrt{400}$$

Whakarūnātia.

$$x-25=20 x-25=-20$$

Me tāpiri 25 ki ngā taha e rua o te whārite.

$$x=45 x=5$$