Whakaoti i te (10-x)(200+50x)=1250

$$(10-x)(200+50x)=1250$$

Whakaoti mō x

$$x=2\sqrt{6}+3\approx 7.898979486$$

$$x=3-2\sqrt{6}\approx -1.898979486$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 10-x ki te 200+50x ka whakakotahi i ngā kupu rite.

$$2000+300x-50x^{2}=1250$$

Tangohia te 1250 mai i ngā taha e rua.

$$2000+300x-50x^{2}-1250=0$$

Tangohia te 1250 i te 2000, ka 750.

$$750+300x-50x^{2}=0$$

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.

$$-50x^{2}+300x+750=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -50 mō a, 300 mō b, me 750 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-300±\sqrt{300^{2}-4\left(-50\right)\times 750}}{2\left(-50\right)}$$

Pūrua 300.

$$x=\frac{-300±\sqrt{90000-4\left(-50\right)\times 750}}{2\left(-50\right)}$$

Whakareatia -4 ki te -50.

$$x=\frac{-300±\sqrt{90000+200\times 750}}{2\left(-50\right)}$$

Whakareatia 200 ki te 750.

$$x=\frac{-300±\sqrt{90000+150000}}{2\left(-50\right)}$$

Tāpiri 90000 ki te 150000.

$$x=\frac{-300±\sqrt{240000}}{2\left(-50\right)}$$

Tuhia te pūtakerua o te 240000.

$$x=\frac{-300±200\sqrt{6}}{2\left(-50\right)}$$

Whakareatia 2 ki te -50.

$$x=\frac{-300±200\sqrt{6}}{-100}$$

Nā, me whakaoti te whārite x=\frac{-300±200\sqrt{6}}{-100} ina he tāpiri te ±. Tāpiri -300 ki te 200\sqrt{6}.

$$x=\frac{200\sqrt{6}-300}{-100}$$

Whakawehe -300+200\sqrt{6} ki te -100.

$$x=3-2\sqrt{6}$$

Nā, me whakaoti te whārite x=\frac{-300±200\sqrt{6}}{-100} ina he tango te ±. Tango 200\sqrt{6} mai i -300.

$$x=\frac{-200\sqrt{6}-300}{-100}$$

Whakawehe -300-200\sqrt{6} ki te -100.

$$x=2\sqrt{6}+3$$

Kua oti te whārite te whakatau.

$$x=3-2\sqrt{6} x=2\sqrt{6}+3$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 10-x ki te 200+50x ka whakakotahi i ngā kupu rite.

$$2000+300x-50x^{2}=1250$$

Tangohia te 2000 mai i ngā taha e rua.

$$300x-50x^{2}=1250-2000$$

Tangohia te 2000 i te 1250, ka -750.

$$300x-50x^{2}=-750$$

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.

$$-50x^{2}+300x=-750$$

Whakawehea ngā taha e rua ki te -50.

$$\frac{-50x^{2}+300x}{-50}=-\frac{750}{-50}$$

Mā te whakawehe ki te -50 ka wetekia te whakareanga ki te -50.

$$x^{2}+\frac{300}{-50}x=-\frac{750}{-50}$$

Whakawehe 300 ki te -50.

$$x^{2}-6x=-\frac{750}{-50}$$

Whakawehe -750 ki te -50.

$$x^{2}-6x=15$$

Whakawehea te -6, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -3. Nā, tāpiria te pūrua o te -3 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-6x+\left(-3\right)^{2}=15+\left(-3\right)^{2}$$

Pūrua -3.

$$x^{2}-6x+9=15+9$$

Tāpiri 15 ki te 9.

$$x^{2}-6x+9=24$$

Tauwehea x^{2}-6x+9. 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}.

$$\left(x-3\right)^{2}=24$$

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

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{24}$$

Whakarūnātia.

$$x-3=2\sqrt{6} x-3=-2\sqrt{6}$$

Me tāpiri 3 ki ngā taha e rua o te whārite.

$$x=2\sqrt{6}+3 x=3-2\sqrt{6}$$