Whakaoti i te 800x+500x(9+x)=6000

$$800x+500x(9+x)=6000$$

Whakaoti mō x

$$x = \frac{\sqrt{4009} - 53}{10} \approx 1.031666447$$

$$x=\frac{-\sqrt{4009}-53}{10}\approx -11.631666447$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 500x ki te 9+x.

$$800x+4500x+500x^{2}=6000$$

Pahekotia te 800x me 4500x, ka 5300x.

$$5300x+500x^{2}=6000$$

Tangohia te 6000 mai i ngā taha e rua.

$$5300x+500x^{2}-6000=0$$

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.

$$500x^{2}+5300x-6000=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 500 mō a, 5300 mō b, me -6000 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5300±\sqrt{5300^{2}-4\times 500\left(-6000\right)}}{2\times 500}$$

Pūrua 5300.

$$x=\frac{-5300±\sqrt{28090000-4\times 500\left(-6000\right)}}{2\times 500}$$

Whakareatia -4 ki te 500.

$$x=\frac{-5300±\sqrt{28090000-2000\left(-6000\right)}}{2\times 500}$$

Whakareatia -2000 ki te -6000.

$$x=\frac{-5300±\sqrt{28090000+12000000}}{2\times 500}$$

Tāpiri 28090000 ki te 12000000.

$$x=\frac{-5300±\sqrt{40090000}}{2\times 500}$$

Tuhia te pūtakerua o te 40090000.

$$x=\frac{-5300±100\sqrt{4009}}{2\times 500}$$

Whakareatia 2 ki te 500.

$$x=\frac{-5300±100\sqrt{4009}}{1000}$$

Nā, me whakaoti te whārite x=\frac{-5300±100\sqrt{4009}}{1000} ina he tāpiri te ±. Tāpiri -5300 ki te 100\sqrt{4009}.

$$x=\frac{100\sqrt{4009}-5300}{1000}$$

Whakawehe -5300+100\sqrt{4009} ki te 1000.

$$x=\frac{\sqrt{4009}-53}{10}$$

Nā, me whakaoti te whārite x=\frac{-5300±100\sqrt{4009}}{1000} ina he tango te ±. Tango 100\sqrt{4009} mai i -5300.

$$x=\frac{-100\sqrt{4009}-5300}{1000}$$

Whakawehe -5300-100\sqrt{4009} ki te 1000.

$$x=\frac{-\sqrt{4009}-53}{10}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{4009}-53}{10} x=\frac{-\sqrt{4009}-53}{10}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 500x ki te 9+x.

$$800x+4500x+500x^{2}=6000$$

Pahekotia te 800x me 4500x, ka 5300x.

$$5300x+500x^{2}=6000$$

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.

$$500x^{2}+5300x=6000$$

Whakawehea ngā taha e rua ki te 500.

$$\frac{500x^{2}+5300x}{500}=\frac{6000}{500}$$

Mā te whakawehe ki te 500 ka wetekia te whakareanga ki te 500.

$$x^{2}+\frac{5300}{500}x=\frac{6000}{500}$$

Whakahekea te hautanga \frac{5300}{500} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 100.

$$x^{2}+\frac{53}{5}x=\frac{6000}{500}$$

Whakawehe 6000 ki te 500.

$$x^{2}+\frac{53}{5}x=12$$

Whakawehea te \frac{53}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{53}{10}. Nā, tāpiria te pūrua o te \frac{53}{10} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{53}{5}x+\left(\frac{53}{10}\right)^{2}=12+\left(\frac{53}{10}\right)^{2}$$

Pūruatia \frac{53}{10} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{53}{5}x+\frac{2809}{100}=12+\frac{2809}{100}$$

Tāpiri 12 ki te \frac{2809}{100}.

$$x^{2}+\frac{53}{5}x+\frac{2809}{100}=\frac{4009}{100}$$

Tauwehea x^{2}+\frac{53}{5}x+\frac{2809}{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}.

$$\left(x+\frac{53}{10}\right)^{2}=\frac{4009}{100}$$

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

$$\sqrt{\left(x+\frac{53}{10}\right)^{2}}=\sqrt{\frac{4009}{100}}$$

Whakarūnātia.

$$x+\frac{53}{10}=\frac{\sqrt{4009}}{10} x+\frac{53}{10}=-\frac{\sqrt{4009}}{10}$$

Me tango \frac{53}{10} mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{4009}-53}{10} x=\frac{-\sqrt{4009}-53}{10}$$