Whakaoti i te (100-x)(1300-5x)=32000

$$(100-x)(1300-5x)=32000$$

Whakaoti mō x

$$x=80\sqrt{2}+180\approx 293.13708499$$

$$x=180-80\sqrt{2}\approx 66.86291501$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 100-x ki te 1300-5x ka whakakotahi i ngā kupu rite.

$$130000-1800x+5x^{2}=32000$$

Tangohia te 32000 mai i ngā taha e rua.

$$130000-1800x+5x^{2}-32000=0$$

Tangohia te 32000 i te 130000, ka 98000.

$$98000-1800x+5x^{2}=0$$

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.

$$5x^{2}-1800x+98000=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 5 mō a, -1800 mō b, me 98000 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1800\right)±\sqrt{\left(-1800\right)^{2}-4\times 5\times 98000}}{2\times 5}$$

Pūrua -1800.

$$x=\frac{-\left(-1800\right)±\sqrt{3240000-4\times 5\times 98000}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$x=\frac{-\left(-1800\right)±\sqrt{3240000-20\times 98000}}{2\times 5}$$

Whakareatia -20 ki te 98000.

$$x=\frac{-\left(-1800\right)±\sqrt{3240000-1960000}}{2\times 5}$$

Tāpiri 3240000 ki te -1960000.

$$x=\frac{-\left(-1800\right)±\sqrt{1280000}}{2\times 5}$$

Tuhia te pūtakerua o te 1280000.

$$x=\frac{-\left(-1800\right)±800\sqrt{2}}{2\times 5}$$

Ko te tauaro o -1800 ko 1800.

$$x=\frac{1800±800\sqrt{2}}{2\times 5}$$

Whakareatia 2 ki te 5.

$$x=\frac{1800±800\sqrt{2}}{10}$$

Nā, me whakaoti te whārite x=\frac{1800±800\sqrt{2}}{10} ina he tāpiri te ±. Tāpiri 1800 ki te 800\sqrt{2}.

$$x=\frac{800\sqrt{2}+1800}{10}$$

Whakawehe 1800+800\sqrt{2} ki te 10.

$$x=80\sqrt{2}+180$$

Nā, me whakaoti te whārite x=\frac{1800±800\sqrt{2}}{10} ina he tango te ±. Tango 800\sqrt{2} mai i 1800.

$$x=\frac{1800-800\sqrt{2}}{10}$$

Whakawehe 1800-800\sqrt{2} ki te 10.

$$x=180-80\sqrt{2}$$

Kua oti te whārite te whakatau.

$$x=80\sqrt{2}+180 x=180-80\sqrt{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 100-x ki te 1300-5x ka whakakotahi i ngā kupu rite.

$$130000-1800x+5x^{2}=32000$$

Tangohia te 130000 mai i ngā taha e rua.

$$-1800x+5x^{2}=32000-130000$$

Tangohia te 130000 i te 32000, ka -98000.

$$-1800x+5x^{2}=-98000$$

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.

$$5x^{2}-1800x=-98000$$

Whakawehea ngā taha e rua ki te 5.

$$\frac{5x^{2}-1800x}{5}=-\frac{98000}{5}$$

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

$$x^{2}+\left(-\frac{1800}{5}\right)x=-\frac{98000}{5}$$

Whakawehe -1800 ki te 5.

$$x^{2}-360x=-\frac{98000}{5}$$

Whakawehe -98000 ki te 5.

$$x^{2}-360x=-19600$$

Whakawehea te -360, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -180. Nā, tāpiria te pūrua o te -180 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-360x+\left(-180\right)^{2}=-19600+\left(-180\right)^{2}$$

Pūrua -180.

$$x^{2}-360x+32400=-19600+32400$$

Tāpiri -19600 ki te 32400.

$$x^{2}-360x+32400=12800$$

Tauwehea x^{2}-360x+32400. 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}.

$$\left(x-180\right)^{2}=12800$$

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

$$\sqrt{\left(x-180\right)^{2}}=\sqrt{12800}$$

Whakarūnātia.

$$x-180=80\sqrt{2} x-180=-80\sqrt{2}$$

Me tāpiri 180 ki ngā taha e rua o te whārite.

$$x=80\sqrt{2}+180 x=180-80\sqrt{2}$$