Whakaoti i te 1302.13=1.22*(1300+x)*x

$$1302.13=1.22 \times (1300+x) \times x$$

Whakaoti mō x

$$x=\frac{\sqrt{6304375986}}{122}-650\approx 0.820497274$$

$$x=-\frac{\sqrt{6304375986}}{122}-650\approx -1300.820497274$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 1.22 ki te 1300+x.

$$1302.13=\left(1586+1.22x\right)x$$

Whakamahia te āhuatanga tohatoha hei whakarea te 1586+1.22x ki te x.

$$1302.13=1586x+1.22x^{2}$$

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

$$1586x+1.22x^{2}=1302.13$$

Tangohia te 1302.13 mai i ngā taha e rua.

$$1586x+1.22x^{2}-1302.13=0$$

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.

$$1.22x^{2}+1586x-1302.13=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1.22 mō a, 1586 mō b, me -1302.13 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1586±\sqrt{1586^{2}-4\times 1.22\left(-1302.13\right)}}{2\times 1.22}$$

Pūrua 1586.

$$x=\frac{-1586±\sqrt{2515396-4\times 1.22\left(-1302.13\right)}}{2\times 1.22}$$

Whakareatia -4 ki te 1.22.

$$x=\frac{-1586±\sqrt{2515396-4.88\left(-1302.13\right)}}{2\times 1.22}$$

Whakareatia -4.88 ki te -1302.13 mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=\frac{-1586±\sqrt{2515396+6354.3944}}{2\times 1.22}$$

Tāpiri 2515396 ki te 6354.3944.

$$x=\frac{-1586±\sqrt{2521750.3944}}{2\times 1.22}$$

Tuhia te pūtakerua o te 2521750.3944.

$$x=\frac{-1586±\frac{\sqrt{6304375986}}{50}}{2\times 1.22}$$

Whakareatia 2 ki te 1.22.

$$x=\frac{-1586±\frac{\sqrt{6304375986}}{50}}{2.44}$$

Nā, me whakaoti te whārite x=\frac{-1586±\frac{\sqrt{6304375986}}{50}}{2.44} ina he tāpiri te ±. Tāpiri -1586 ki te \frac{\sqrt{6304375986}}{50}.

$$x=\frac{\frac{\sqrt{6304375986}}{50}-1586}{2.44}$$

Whakawehe -1586+\frac{\sqrt{6304375986}}{50} ki te 2.44 mā te whakarea -1586+\frac{\sqrt{6304375986}}{50} ki te tau huripoki o 2.44.

$$x=\frac{\sqrt{6304375986}}{122}-650$$

Nā, me whakaoti te whārite x=\frac{-1586±\frac{\sqrt{6304375986}}{50}}{2.44} ina he tango te ±. Tango \frac{\sqrt{6304375986}}{50} mai i -1586.

$$x=\frac{-\frac{\sqrt{6304375986}}{50}-1586}{2.44}$$

Whakawehe -1586-\frac{\sqrt{6304375986}}{50} ki te 2.44 mā te whakarea -1586-\frac{\sqrt{6304375986}}{50} ki te tau huripoki o 2.44.

$$x=-\frac{\sqrt{6304375986}}{122}-650$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{6304375986}}{122}-650 x=-\frac{\sqrt{6304375986}}{122}-650$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 1.22 ki te 1300+x.

$$1302.13=\left(1586+1.22x\right)x$$

Whakamahia te āhuatanga tohatoha hei whakarea te 1586+1.22x ki te x.

$$1302.13=1586x+1.22x^{2}$$

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

$$1586x+1.22x^{2}=1302.13$$

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.

$$1.22x^{2}+1586x=1302.13$$

Whakawehea ngā taha e rua o te whārite ki te 1.22, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$\frac{1.22x^{2}+1586x}{1.22}=\frac{1302.13}{1.22}$$

Mā te whakawehe ki te 1.22 ka wetekia te whakareanga ki te 1.22.

$$x^{2}+\frac{1586}{1.22}x=\frac{1302.13}{1.22}$$

Whakawehe 1586 ki te 1.22 mā te whakarea 1586 ki te tau huripoki o 1.22.

$$x^{2}+1300x=\frac{1302.13}{1.22}$$

Whakawehe 1302.13 ki te 1.22 mā te whakarea 1302.13 ki te tau huripoki o 1.22.

$$x^{2}+1300x=\frac{130213}{122}$$

Whakawehea te 1300, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 650. Nā, tāpiria te pūrua o te 650 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+1300x+650^{2}=\frac{130213}{122}+650^{2}$$

Pūrua 650.

$$x^{2}+1300x+422500=\frac{130213}{122}+422500$$

Tāpiri \frac{130213}{122} ki te 422500.

$$x^{2}+1300x+422500=\frac{51675213}{122}$$

Tauwehea x^{2}+1300x+422500. 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}.

$$\left(x+650\right)^{2}=\frac{51675213}{122}$$

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

$$\sqrt{\left(x+650\right)^{2}}=\sqrt{\frac{51675213}{122}}$$

Whakarūnātia.

$$x+650=\frac{\sqrt{6304375986}}{122} x+650=-\frac{\sqrt{6304375986}}{122}$$

Me tango 650 mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{6304375986}}{122}-650 x=-\frac{\sqrt{6304375986}}{122}-650$$