Whakaoti i te 100+400=1600+x^2-80x

$$100+400=1600+ { x }^{ 2 } -80x$$

Whakaoti mō x

$$x=10\sqrt{5}+40\approx 62.360679775$$

$$x=40-10\sqrt{5}\approx 17.639320225$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tāpirihia te 100 ki te 400, ka 500.

$$500=1600+x^{2}-80x$$

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

$$1600+x^{2}-80x=500$$

Tangohia te 500 mai i ngā taha e rua.

$$1600+x^{2}-80x-500=0$$

Tangohia te 500 i te 1600, ka 1100.

$$1100+x^{2}-80x=0$$

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.

$$x^{2}-80x+1100=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -80 mō b, me 1100 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-80\right)±\sqrt{\left(-80\right)^{2}-4\times 1100}}{2}$$

Pūrua -80.

$$x=\frac{-\left(-80\right)±\sqrt{6400-4\times 1100}}{2}$$

Whakareatia -4 ki te 1100.

$$x=\frac{-\left(-80\right)±\sqrt{6400-4400}}{2}$$

Tāpiri 6400 ki te -4400.

$$x=\frac{-\left(-80\right)±\sqrt{2000}}{2}$$

Tuhia te pūtakerua o te 2000.

$$x=\frac{-\left(-80\right)±20\sqrt{5}}{2}$$

Ko te tauaro o -80 ko 80.

$$x=\frac{80±20\sqrt{5}}{2}$$

Nā, me whakaoti te whārite x=\frac{80±20\sqrt{5}}{2} ina he tāpiri te ±. Tāpiri 80 ki te 20\sqrt{5}.

$$x=\frac{20\sqrt{5}+80}{2}$$

Whakawehe 80+20\sqrt{5} ki te 2.

$$x=10\sqrt{5}+40$$

Nā, me whakaoti te whārite x=\frac{80±20\sqrt{5}}{2} ina he tango te ±. Tango 20\sqrt{5} mai i 80.

$$x=\frac{80-20\sqrt{5}}{2}$$

Whakawehe 80-20\sqrt{5} ki te 2.

$$x=40-10\sqrt{5}$$

Kua oti te whārite te whakatau.

$$x=10\sqrt{5}+40 x=40-10\sqrt{5}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tāpirihia te 100 ki te 400, ka 500.

$$500=1600+x^{2}-80x$$

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

$$1600+x^{2}-80x=500$$

Tangohia te 1600 mai i ngā taha e rua.

$$x^{2}-80x=500-1600$$

Tangohia te 1600 i te 500, ka -1100.

$$x^{2}-80x=-1100$$

Whakawehea te -80, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -40. Nā, tāpiria te pūrua o te -40 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-80x+\left(-40\right)^{2}=-1100+\left(-40\right)^{2}$$

Pūrua -40.

$$x^{2}-80x+1600=-1100+1600$$

Tāpiri -1100 ki te 1600.

$$x^{2}-80x+1600=500$$

Tauwehea te x^{2}-80x+1600. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-40\right)^{2}=500$$

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

$$\sqrt{\left(x-40\right)^{2}}=\sqrt{500}$$

Whakarūnātia.

$$x-40=10\sqrt{5} x-40=-10\sqrt{5}$$

Me tāpiri 40 ki ngā taha e rua o te whārite.

$$x=10\sqrt{5}+40 x=40-10\sqrt{5}$$