Whakaoti i te +(1+5x)(100-x)=10

$$+ ( 1 + 5 x ) ( 100 - x ) = 10$$

Whakaoti mō x

$$x = \frac{\sqrt{250801} + 499}{10} \approx 99.980035942$$

$$x=\frac{499-\sqrt{250801}}{10}\approx -0.180035942$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 1+5x ki te 100-x ka whakakotahi i ngā kupu rite.

$$100+499x-5x^{2}=10$$

Tangohia te 10 mai i ngā taha e rua.

$$100+499x-5x^{2}-10=0$$

Tangohia te 10 i te 100, ka 90.

$$90+499x-5x^{2}=0$$

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.

$$-5x^{2}+499x+90=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -5 mō a, 499 mō b, me 90 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-499±\sqrt{499^{2}-4\left(-5\right)\times 90}}{2\left(-5\right)}$$

Pūrua 499.

$$x=\frac{-499±\sqrt{249001-4\left(-5\right)\times 90}}{2\left(-5\right)}$$

Whakareatia -4 ki te -5.

$$x=\frac{-499±\sqrt{249001+20\times 90}}{2\left(-5\right)}$$

Whakareatia 20 ki te 90.

$$x=\frac{-499±\sqrt{249001+1800}}{2\left(-5\right)}$$

Tāpiri 249001 ki te 1800.

$$x=\frac{-499±\sqrt{250801}}{2\left(-5\right)}$$

Whakareatia 2 ki te -5.

$$x=\frac{-499±\sqrt{250801}}{-10}$$

Nā, me whakaoti te whārite x=\frac{-499±\sqrt{250801}}{-10} ina he tāpiri te ±. Tāpiri -499 ki te \sqrt{250801}.

$$x=\frac{\sqrt{250801}-499}{-10}$$

Whakawehe -499+\sqrt{250801} ki te -10.

$$x=\frac{499-\sqrt{250801}}{10}$$

Nā, me whakaoti te whārite x=\frac{-499±\sqrt{250801}}{-10} ina he tango te ±. Tango \sqrt{250801} mai i -499.

$$x=\frac{-\sqrt{250801}-499}{-10}$$

Whakawehe -499-\sqrt{250801} ki te -10.

$$x=\frac{\sqrt{250801}+499}{10}$$

Kua oti te whārite te whakatau.

$$x=\frac{499-\sqrt{250801}}{10} x=\frac{\sqrt{250801}+499}{10}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 1+5x ki te 100-x ka whakakotahi i ngā kupu rite.

$$100+499x-5x^{2}=10$$

Tangohia te 100 mai i ngā taha e rua.

$$499x-5x^{2}=10-100$$

Tangohia te 100 i te 10, ka -90.

$$499x-5x^{2}=-90$$

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.

$$-5x^{2}+499x=-90$$

Whakawehea ngā taha e rua ki te -5.

$$\frac{-5x^{2}+499x}{-5}=-\frac{90}{-5}$$

Mā te whakawehe ki te -5 ka wetekia te whakareanga ki te -5.

$$x^{2}+\frac{499}{-5}x=-\frac{90}{-5}$$

Whakawehe 499 ki te -5.

$$x^{2}-\frac{499}{5}x=-\frac{90}{-5}$$

Whakawehe -90 ki te -5.

$$x^{2}-\frac{499}{5}x=18$$

Whakawehea te -\frac{499}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{499}{10}. Nā, tāpiria te pūrua o te -\frac{499}{10} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{499}{5}x+\left(-\frac{499}{10}\right)^{2}=18+\left(-\frac{499}{10}\right)^{2}$$

Pūruatia -\frac{499}{10} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{499}{5}x+\frac{249001}{100}=18+\frac{249001}{100}$$

Tāpiri 18 ki te \frac{249001}{100}.

$$x^{2}-\frac{499}{5}x+\frac{249001}{100}=\frac{250801}{100}$$

Tauwehea x^{2}-\frac{499}{5}x+\frac{249001}{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}.

$$\left(x-\frac{499}{10}\right)^{2}=\frac{250801}{100}$$

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

$$\sqrt{\left(x-\frac{499}{10}\right)^{2}}=\sqrt{\frac{250801}{100}}$$

Whakarūnātia.

$$x-\frac{499}{10}=\frac{\sqrt{250801}}{10} x-\frac{499}{10}=-\frac{\sqrt{250801}}{10}$$

Me tāpiri \frac{499}{10} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{250801}+499}{10} x=\frac{499-\sqrt{250801}}{10}$$